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Exact Named Norms, Universal Certificates and Induced Distances

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batch-20261013-advance-20261026-r02-real-normed-vector-space-foundations · 2026-10-13 · 100 questions

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

1. Evaluate a Named l1 Norm Exactly

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[-12, -8, -9, -2]. Evaluate l1(v). Give (vector,norm_name,value).

View answer and explanation

([-12, -8, -9, -2], l1, 31)

Sum the coordinate absolute values in the named formula; preserve the original signed vector. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-12, -8, -9, -2], l1, 31).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-l1-evaluate · candidate

2. Evaluate a Named l1 Norm Exactly

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[-11, -7, -2, -1]. Evaluate l1(v). Give (vector,norm_name,value).

View answer and explanation

([-11, -7, -2, -1], l1, 21)

Sum the coordinate absolute values in the named formula; preserve the original signed vector. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-11, -7, -2, -1], l1, 21).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-l1-evaluate · candidate

3. Evaluate a Named l1 Norm Exactly

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[-10, -4, 5, 0]. Evaluate l1(v). Give (vector,norm_name,value).

View answer and explanation

([-10, -4, 5, 0], l1, 19)

Sum the coordinate absolute values in the named formula; preserve the original signed vector. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-10, -4, 5, 0], l1, 19).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-l1-evaluate · candidate

4. Evaluate a Named l1 Norm Exactly

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[-9, 1, -7, 1]. Evaluate l1(v). Give (vector,norm_name,value).

View answer and explanation

([-9, 1, -7, 1], l1, 18)

Sum the coordinate absolute values in the named formula; preserve the original signed vector. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-9, 1, -7, 1], l1, 18).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-l1-evaluate · candidate

5. Evaluate a Named l1 Norm Exactly

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[-8, 8, 0, 2]. Evaluate l1(v). Give (vector,norm_name,value).

View answer and explanation

([-8, 8, 0, 2], l1, 18)

Sum the coordinate absolute values in the named formula; preserve the original signed vector. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-8, 8, 0, 2], l1, 18).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-l1-evaluate · candidate

6. Evaluate a Named l1 Norm Exactly

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[-7, 0, 7, -2]. Evaluate l1(v). Give (vector,norm_name,value).

View answer and explanation

([-7, 0, 7, -2], l1, 16)

Sum the coordinate absolute values in the named formula; preserve the original signed vector. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-7, 0, 7, -2], l1, 16).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-l1-evaluate · candidate

7. Evaluate a Named l1 Norm Exactly

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[-6, -6, -5, -1]. Evaluate l1(v). Give (vector,norm_name,value).

View answer and explanation

([-6, -6, -5, -1], l1, 18)

Sum the coordinate absolute values in the named formula; preserve the original signed vector. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-6, -6, -5, -1], l1, 18).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-l1-evaluate · candidate

8. Evaluate a Named l1 Norm Exactly

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[-5, 7, 2, 0]. Evaluate l1(v). Give (vector,norm_name,value).

View answer and explanation

([-5, 7, 2, 0], l1, 14)

Sum the coordinate absolute values in the named formula; preserve the original signed vector. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-5, 7, 2, 0], l1, 14).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-l1-evaluate · candidate

9. Evaluate a Named l1 Norm Exactly

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[-4, 5, 9, 1]. Evaluate l1(v). Give (vector,norm_name,value).

View answer and explanation

([-4, 5, 9, 1], l1, 19)

Sum the coordinate absolute values in the named formula; preserve the original signed vector. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-4, 5, 9, 1], l1, 19).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-l1-evaluate · candidate

10. Evaluate a Named l1 Norm Exactly

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[-3, 5, -3, 2]. Evaluate l1(v). Give (vector,norm_name,value).

View answer and explanation

([-3, 5, -3, 2], l1, 13)

Sum the coordinate absolute values in the named formula; preserve the original signed vector. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-3, 5, -3, 2], l1, 13).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-l1-evaluate · candidate

11. Evaluate a Named l1 Norm Exactly

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[-2, 7, 4, -2]. Evaluate l1(v). Give (vector,norm_name,value).

View answer and explanation

([-2, 7, 4, -2], l1, 15)

Sum the coordinate absolute values in the named formula; preserve the original signed vector. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-2, 7, 4, -2], l1, 15).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-l1-evaluate · candidate

12. Evaluate a Named l1 Norm Exactly

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[-1, -6, -8, -1]. Evaluate l1(v). Give (vector,norm_name,value).

View answer and explanation

([-1, -6, -8, -1], l1, 16)

Sum the coordinate absolute values in the named formula; preserve the original signed vector. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-1, -6, -8, -1], l1, 16).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-l1-evaluate · candidate

13. Evaluate a Named l1 Norm Exactly

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[0, 0, -1, 0]. Evaluate l1(v). Give (vector,norm_name,value).

View answer and explanation

([0, 0, -1, 0], l1, 1)

Sum the coordinate absolute values in the named formula; preserve the original signed vector. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([0, 0, -1, 0], l1, 1).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-l1-evaluate · candidate

14. Evaluate a Named l1 Norm Exactly

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[1, 8, 6, 1]. Evaluate l1(v). Give (vector,norm_name,value).

View answer and explanation

([1, 8, 6, 1], l1, 16)

Sum the coordinate absolute values in the named formula; preserve the original signed vector. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([1, 8, 6, 1], l1, 16).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-l1-evaluate · candidate

15. Evaluate a Named l1 Norm Exactly

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[2, 1, -6, 2]. Evaluate l1(v). Give (vector,norm_name,value).

View answer and explanation

([2, 1, -6, 2], l1, 11)

Sum the coordinate absolute values in the named formula; preserve the original signed vector. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([2, 1, -6, 2], l1, 11).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-l1-evaluate · candidate

16. Evaluate a Named l1 Norm Exactly

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[3, -4, 1, -2]. Evaluate l1(v). Give (vector,norm_name,value).

View answer and explanation

([3, -4, 1, -2], l1, 10)

Sum the coordinate absolute values in the named formula; preserve the original signed vector. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([3, -4, 1, -2], l1, 10).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-l1-evaluate · candidate

17. Evaluate a Named l1 Norm Exactly

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[4, -7, 8, -1]. Evaluate l1(v). Give (vector,norm_name,value).

View answer and explanation

([4, -7, 8, -1], l1, 20)

Sum the coordinate absolute values in the named formula; preserve the original signed vector. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([4, -7, 8, -1], l1, 20).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-l1-evaluate · candidate

18. Evaluate a Named l1 Norm Exactly

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[5, -8, -4, 0]. Evaluate l1(v). Give (vector,norm_name,value).

View answer and explanation

([5, -8, -4, 0], l1, 17)

Sum the coordinate absolute values in the named formula; preserve the original signed vector. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([5, -8, -4, 0], l1, 17).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-l1-evaluate · candidate

19. Evaluate a Named l1 Norm Exactly

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[6, -7, 3, 1]. Evaluate l1(v). Give (vector,norm_name,value).

View answer and explanation

([6, -7, 3, 1], l1, 17)

Sum the coordinate absolute values in the named formula; preserve the original signed vector. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([6, -7, 3, 1], l1, 17).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-l1-evaluate · candidate

20. Evaluate a Named l1 Norm Exactly

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[7, -4, -9, 2]. Evaluate l1(v). Give (vector,norm_name,value).

View answer and explanation

([7, -4, -9, 2], l1, 22)

Sum the coordinate absolute values in the named formula; preserve the original signed vector. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([7, -4, -9, 2], l1, 22).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-l1-evaluate · candidate

21. Evaluate a Named l1 Norm Exactly

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[8, 1, -2, -2]. Evaluate l1(v). Give (vector,norm_name,value).

View answer and explanation

([8, 1, -2, -2], l1, 13)

Sum the coordinate absolute values in the named formula; preserve the original signed vector. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([8, 1, -2, -2], l1, 13).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-l1-evaluate · candidate

22. Evaluate a Named l1 Norm Exactly

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[9, 8, 5, -1]. Evaluate l1(v). Give (vector,norm_name,value).

View answer and explanation

([9, 8, 5, -1], l1, 23)

Sum the coordinate absolute values in the named formula; preserve the original signed vector. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([9, 8, 5, -1], l1, 23).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-l1-evaluate · candidate

23. Evaluate a Named l1 Norm Exactly

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[10, 0, -7, 0]. Evaluate l1(v). Give (vector,norm_name,value).

View answer and explanation

([10, 0, -7, 0], l1, 17)

Sum the coordinate absolute values in the named formula; preserve the original signed vector. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([10, 0, -7, 0], l1, 17).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-l1-evaluate · candidate

24. Evaluate a Named l1 Norm Exactly

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[11, -6, 0, 1]. Evaluate l1(v). Give (vector,norm_name,value).

View answer and explanation

([11, -6, 0, 1], l1, 18)

Sum the coordinate absolute values in the named formula; preserve the original signed vector. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([11, -6, 0, 1], l1, 18).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-l1-evaluate · candidate

25. Evaluate a Named l1 Norm Exactly

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[12, 7, 7, 2]. Evaluate l1(v). Give (vector,norm_name,value).

View answer and explanation

([12, 7, 7, 2], l1, 28)

Sum the coordinate absolute values in the named formula; preserve the original signed vector. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([12, 7, 7, 2], l1, 28).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-l1-evaluate · candidate

26. Evaluate a Named Maximum Norm Exactly

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[-12, -8, -9, -2]. Evaluate linfinity(v). Give (vector,norm_name,value).

View answer and explanation

([-12, -8, -9, -2], linfinity, 12)

Take the maximum coordinate absolute value, not the largest signed coordinate. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-12, -8, -9, -2], linfinity, 12).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-linfinity-evaluate · candidate

27. Evaluate a Named Maximum Norm Exactly

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[-11, -7, -2, -1]. Evaluate linfinity(v). Give (vector,norm_name,value).

View answer and explanation

([-11, -7, -2, -1], linfinity, 11)

Take the maximum coordinate absolute value, not the largest signed coordinate. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-11, -7, -2, -1], linfinity, 11).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-linfinity-evaluate · candidate

28. Evaluate a Named Maximum Norm Exactly

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[-10, -4, 5, 0]. Evaluate linfinity(v). Give (vector,norm_name,value).

View answer and explanation

([-10, -4, 5, 0], linfinity, 10)

Take the maximum coordinate absolute value, not the largest signed coordinate. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-10, -4, 5, 0], linfinity, 10).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-linfinity-evaluate · candidate

29. Evaluate a Named Maximum Norm Exactly

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[-9, 1, -7, 1]. Evaluate linfinity(v). Give (vector,norm_name,value).

View answer and explanation

([-9, 1, -7, 1], linfinity, 9)

Take the maximum coordinate absolute value, not the largest signed coordinate. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-9, 1, -7, 1], linfinity, 9).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-linfinity-evaluate · candidate

30. Evaluate a Named Maximum Norm Exactly

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[-8, 8, 0, 2]. Evaluate linfinity(v). Give (vector,norm_name,value).

View answer and explanation

([-8, 8, 0, 2], linfinity, 8)

Take the maximum coordinate absolute value, not the largest signed coordinate. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-8, 8, 0, 2], linfinity, 8).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-linfinity-evaluate · candidate

31. Evaluate a Named Maximum Norm Exactly

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[-7, 0, 7, -2]. Evaluate linfinity(v). Give (vector,norm_name,value).

View answer and explanation

([-7, 0, 7, -2], linfinity, 7)

Take the maximum coordinate absolute value, not the largest signed coordinate. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-7, 0, 7, -2], linfinity, 7).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-linfinity-evaluate · candidate

32. Evaluate a Named Maximum Norm Exactly

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[-6, -6, -5, -1]. Evaluate linfinity(v). Give (vector,norm_name,value).

View answer and explanation

([-6, -6, -5, -1], linfinity, 6)

Take the maximum coordinate absolute value, not the largest signed coordinate. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-6, -6, -5, -1], linfinity, 6).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-linfinity-evaluate · candidate

33. Evaluate a Named Maximum Norm Exactly

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[-5, 7, 2, 0]. Evaluate linfinity(v). Give (vector,norm_name,value).

View answer and explanation

([-5, 7, 2, 0], linfinity, 7)

Take the maximum coordinate absolute value, not the largest signed coordinate. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-5, 7, 2, 0], linfinity, 7).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-linfinity-evaluate · candidate

34. Evaluate a Named Maximum Norm Exactly

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[-4, 5, 9, 1]. Evaluate linfinity(v). Give (vector,norm_name,value).

View answer and explanation

([-4, 5, 9, 1], linfinity, 9)

Take the maximum coordinate absolute value, not the largest signed coordinate. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-4, 5, 9, 1], linfinity, 9).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-linfinity-evaluate · candidate

35. Evaluate a Named Maximum Norm Exactly

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[-3, 5, -3, 2]. Evaluate linfinity(v). Give (vector,norm_name,value).

View answer and explanation

([-3, 5, -3, 2], linfinity, 5)

Take the maximum coordinate absolute value, not the largest signed coordinate. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-3, 5, -3, 2], linfinity, 5).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-linfinity-evaluate · candidate

36. Evaluate a Named Maximum Norm Exactly

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[-2, 7, 4, -2]. Evaluate linfinity(v). Give (vector,norm_name,value).

View answer and explanation

([-2, 7, 4, -2], linfinity, 7)

Take the maximum coordinate absolute value, not the largest signed coordinate. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-2, 7, 4, -2], linfinity, 7).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-linfinity-evaluate · candidate

37. Evaluate a Named Maximum Norm Exactly

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[-1, -6, -8, -1]. Evaluate linfinity(v). Give (vector,norm_name,value).

View answer and explanation

([-1, -6, -8, -1], linfinity, 8)

Take the maximum coordinate absolute value, not the largest signed coordinate. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([-1, -6, -8, -1], linfinity, 8).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-linfinity-evaluate · candidate

38. Evaluate a Named Maximum Norm Exactly

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[0, 0, -1, 0]. Evaluate linfinity(v). Give (vector,norm_name,value).

View answer and explanation

([0, 0, -1, 0], linfinity, 1)

Take the maximum coordinate absolute value, not the largest signed coordinate. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([0, 0, -1, 0], linfinity, 1).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-linfinity-evaluate · candidate

39. Evaluate a Named Maximum Norm Exactly

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[1, 8, 6, 1]. Evaluate linfinity(v). Give (vector,norm_name,value).

View answer and explanation

([1, 8, 6, 1], linfinity, 8)

Take the maximum coordinate absolute value, not the largest signed coordinate. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([1, 8, 6, 1], linfinity, 8).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-linfinity-evaluate · candidate

40. Evaluate a Named Maximum Norm Exactly

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[2, 1, -6, 2]. Evaluate linfinity(v). Give (vector,norm_name,value).

View answer and explanation

([2, 1, -6, 2], linfinity, 6)

Take the maximum coordinate absolute value, not the largest signed coordinate. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([2, 1, -6, 2], linfinity, 6).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-linfinity-evaluate · candidate

41. Evaluate a Named Maximum Norm Exactly

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[3, -4, 1, -2]. Evaluate linfinity(v). Give (vector,norm_name,value).

View answer and explanation

([3, -4, 1, -2], linfinity, 4)

Take the maximum coordinate absolute value, not the largest signed coordinate. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([3, -4, 1, -2], linfinity, 4).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-linfinity-evaluate · candidate

42. Evaluate a Named Maximum Norm Exactly

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[4, -7, 8, -1]. Evaluate linfinity(v). Give (vector,norm_name,value).

View answer and explanation

([4, -7, 8, -1], linfinity, 8)

Take the maximum coordinate absolute value, not the largest signed coordinate. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([4, -7, 8, -1], linfinity, 8).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-linfinity-evaluate · candidate

43. Evaluate a Named Maximum Norm Exactly

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[5, -8, -4, 0]. Evaluate linfinity(v). Give (vector,norm_name,value).

View answer and explanation

([5, -8, -4, 0], linfinity, 8)

Take the maximum coordinate absolute value, not the largest signed coordinate. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([5, -8, -4, 0], linfinity, 8).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-linfinity-evaluate · candidate

44. Evaluate a Named Maximum Norm Exactly

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[6, -7, 3, 1]. Evaluate linfinity(v). Give (vector,norm_name,value).

View answer and explanation

([6, -7, 3, 1], linfinity, 7)

Take the maximum coordinate absolute value, not the largest signed coordinate. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([6, -7, 3, 1], linfinity, 7).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-linfinity-evaluate · candidate

45. Evaluate a Named Maximum Norm Exactly

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[7, -4, -9, 2]. Evaluate linfinity(v). Give (vector,norm_name,value).

View answer and explanation

([7, -4, -9, 2], linfinity, 9)

Take the maximum coordinate absolute value, not the largest signed coordinate. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([7, -4, -9, 2], linfinity, 9).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-linfinity-evaluate · candidate

46. Evaluate a Named Maximum Norm Exactly

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[8, 1, -2, -2]. Evaluate linfinity(v). Give (vector,norm_name,value).

View answer and explanation

([8, 1, -2, -2], linfinity, 8)

Take the maximum coordinate absolute value, not the largest signed coordinate. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([8, 1, -2, -2], linfinity, 8).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-linfinity-evaluate · candidate

47. Evaluate a Named Maximum Norm Exactly

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[9, 8, 5, -1]. Evaluate linfinity(v). Give (vector,norm_name,value).

View answer and explanation

([9, 8, 5, -1], linfinity, 9)

Take the maximum coordinate absolute value, not the largest signed coordinate. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([9, 8, 5, -1], linfinity, 9).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-linfinity-evaluate · candidate

48. Evaluate a Named Maximum Norm Exactly

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[10, 0, -7, 0]. Evaluate linfinity(v). Give (vector,norm_name,value).

View answer and explanation

([10, 0, -7, 0], linfinity, 10)

Take the maximum coordinate absolute value, not the largest signed coordinate. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([10, 0, -7, 0], linfinity, 10).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-linfinity-evaluate · candidate

49. Evaluate a Named Maximum Norm Exactly

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[11, -6, 0, 1]. Evaluate linfinity(v). Give (vector,norm_name,value).

View answer and explanation

([11, -6, 0, 1], linfinity, 11)

Take the maximum coordinate absolute value, not the largest signed coordinate. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([11, -6, 0, 1], linfinity, 11).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-linfinity-evaluate · candidate

50. Evaluate a Named Maximum Norm Exactly

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. Integer vector=[12, 7, 7, 2]. Evaluate linfinity(v). Give (vector,norm_name,value).

View answer and explanation

([12, 7, 7, 2], linfinity, 12)

Take the maximum coordinate absolute value, not the largest signed coordinate. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([12, 7, 7, 2], linfinity, 12).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-linfinity-evaluate · candidate

51. A Named-Formula Certificate Is Universal, Not Sampled

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=l1, dimension=1. Give (norm_name,dimension,qualified?,positivity+absolute_homogeneity+triangle_inequality_by_named_formula,not_sampled).

View answer and explanation

(l1, 1, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled)

The absolute-value identities and inequalities prove positivity, absolute homogeneity and the triangle inequality for all vectors and real scalars in this dimension. No sampled vectors establish the axioms. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 1, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-named-norm-certificate · candidate

52. A Named-Formula Certificate Is Universal, Not Sampled

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=linfinity, dimension=2. Give (norm_name,dimension,qualified?,positivity+absolute_homogeneity+triangle_inequality_by_named_formula,not_sampled).

View answer and explanation

(linfinity, 2, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled)

The absolute-value identities and inequalities prove positivity, absolute homogeneity and the triangle inequality for all vectors and real scalars in this dimension. No sampled vectors establish the axioms. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 2, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-named-norm-certificate · candidate

53. A Named-Formula Certificate Is Universal, Not Sampled

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=l1, dimension=3. Give (norm_name,dimension,qualified?,positivity+absolute_homogeneity+triangle_inequality_by_named_formula,not_sampled).

View answer and explanation

(l1, 3, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled)

The absolute-value identities and inequalities prove positivity, absolute homogeneity and the triangle inequality for all vectors and real scalars in this dimension. No sampled vectors establish the axioms. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 3, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-named-norm-certificate · candidate

54. A Named-Formula Certificate Is Universal, Not Sampled

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=linfinity, dimension=4. Give (norm_name,dimension,qualified?,positivity+absolute_homogeneity+triangle_inequality_by_named_formula,not_sampled).

View answer and explanation

(linfinity, 4, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled)

The absolute-value identities and inequalities prove positivity, absolute homogeneity and the triangle inequality for all vectors and real scalars in this dimension. No sampled vectors establish the axioms. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 4, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-named-norm-certificate · candidate

55. A Named-Formula Certificate Is Universal, Not Sampled

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=l1, dimension=5. Give (norm_name,dimension,qualified?,positivity+absolute_homogeneity+triangle_inequality_by_named_formula,not_sampled).

View answer and explanation

(l1, 5, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled)

The absolute-value identities and inequalities prove positivity, absolute homogeneity and the triangle inequality for all vectors and real scalars in this dimension. No sampled vectors establish the axioms. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 5, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-named-norm-certificate · candidate

56. A Named-Formula Certificate Is Universal, Not Sampled

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=linfinity, dimension=6. Give (norm_name,dimension,qualified?,positivity+absolute_homogeneity+triangle_inequality_by_named_formula,not_sampled).

View answer and explanation

(linfinity, 6, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled)

The absolute-value identities and inequalities prove positivity, absolute homogeneity and the triangle inequality for all vectors and real scalars in this dimension. No sampled vectors establish the axioms. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 6, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-named-norm-certificate · candidate

57. A Named-Formula Certificate Is Universal, Not Sampled

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=l1, dimension=7. Give (norm_name,dimension,qualified?,positivity+absolute_homogeneity+triangle_inequality_by_named_formula,not_sampled).

View answer and explanation

(l1, 7, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled)

The absolute-value identities and inequalities prove positivity, absolute homogeneity and the triangle inequality for all vectors and real scalars in this dimension. No sampled vectors establish the axioms. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 7, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-named-norm-certificate · candidate

58. A Named-Formula Certificate Is Universal, Not Sampled

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=linfinity, dimension=8. Give (norm_name,dimension,qualified?,positivity+absolute_homogeneity+triangle_inequality_by_named_formula,not_sampled).

View answer and explanation

(linfinity, 8, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled)

The absolute-value identities and inequalities prove positivity, absolute homogeneity and the triangle inequality for all vectors and real scalars in this dimension. No sampled vectors establish the axioms. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 8, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-named-norm-certificate · candidate

59. A Named-Formula Certificate Is Universal, Not Sampled

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=l1, dimension=9. Give (norm_name,dimension,qualified?,positivity+absolute_homogeneity+triangle_inequality_by_named_formula,not_sampled).

View answer and explanation

(l1, 9, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled)

The absolute-value identities and inequalities prove positivity, absolute homogeneity and the triangle inequality for all vectors and real scalars in this dimension. No sampled vectors establish the axioms. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 9, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-named-norm-certificate · candidate

60. A Named-Formula Certificate Is Universal, Not Sampled

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=linfinity, dimension=10. Give (norm_name,dimension,qualified?,positivity+absolute_homogeneity+triangle_inequality_by_named_formula,not_sampled).

View answer and explanation

(linfinity, 10, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled)

The absolute-value identities and inequalities prove positivity, absolute homogeneity and the triangle inequality for all vectors and real scalars in this dimension. No sampled vectors establish the axioms. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 10, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-named-norm-certificate · candidate

61. A Named-Formula Certificate Is Universal, Not Sampled

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=l1, dimension=11. Give (norm_name,dimension,qualified?,positivity+absolute_homogeneity+triangle_inequality_by_named_formula,not_sampled).

View answer and explanation

(l1, 11, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled)

The absolute-value identities and inequalities prove positivity, absolute homogeneity and the triangle inequality for all vectors and real scalars in this dimension. No sampled vectors establish the axioms. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 11, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-named-norm-certificate · candidate

62. A Named-Formula Certificate Is Universal, Not Sampled

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=linfinity, dimension=12. Give (norm_name,dimension,qualified?,positivity+absolute_homogeneity+triangle_inequality_by_named_formula,not_sampled).

View answer and explanation

(linfinity, 12, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled)

The absolute-value identities and inequalities prove positivity, absolute homogeneity and the triangle inequality for all vectors and real scalars in this dimension. No sampled vectors establish the axioms. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 12, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-named-norm-certificate · candidate

63. A Named-Formula Certificate Is Universal, Not Sampled

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=l1, dimension=13. Give (norm_name,dimension,qualified?,positivity+absolute_homogeneity+triangle_inequality_by_named_formula,not_sampled).

View answer and explanation

(l1, 13, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled)

The absolute-value identities and inequalities prove positivity, absolute homogeneity and the triangle inequality for all vectors and real scalars in this dimension. No sampled vectors establish the axioms. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 13, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-named-norm-certificate · candidate

64. A Named-Formula Certificate Is Universal, Not Sampled

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=linfinity, dimension=14. Give (norm_name,dimension,qualified?,positivity+absolute_homogeneity+triangle_inequality_by_named_formula,not_sampled).

View answer and explanation

(linfinity, 14, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled)

The absolute-value identities and inequalities prove positivity, absolute homogeneity and the triangle inequality for all vectors and real scalars in this dimension. No sampled vectors establish the axioms. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 14, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-named-norm-certificate · candidate

65. A Named-Formula Certificate Is Universal, Not Sampled

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=l1, dimension=15. Give (norm_name,dimension,qualified?,positivity+absolute_homogeneity+triangle_inequality_by_named_formula,not_sampled).

View answer and explanation

(l1, 15, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled)

The absolute-value identities and inequalities prove positivity, absolute homogeneity and the triangle inequality for all vectors and real scalars in this dimension. No sampled vectors establish the axioms. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 15, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-named-norm-certificate · candidate

66. A Named-Formula Certificate Is Universal, Not Sampled

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=linfinity, dimension=16. Give (norm_name,dimension,qualified?,positivity+absolute_homogeneity+triangle_inequality_by_named_formula,not_sampled).

View answer and explanation

(linfinity, 16, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled)

The absolute-value identities and inequalities prove positivity, absolute homogeneity and the triangle inequality for all vectors and real scalars in this dimension. No sampled vectors establish the axioms. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 16, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-named-norm-certificate · candidate

67. A Named-Formula Certificate Is Universal, Not Sampled

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=l1, dimension=17. Give (norm_name,dimension,qualified?,positivity+absolute_homogeneity+triangle_inequality_by_named_formula,not_sampled).

View answer and explanation

(l1, 17, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled)

The absolute-value identities and inequalities prove positivity, absolute homogeneity and the triangle inequality for all vectors and real scalars in this dimension. No sampled vectors establish the axioms. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 17, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-named-norm-certificate · candidate

68. A Named-Formula Certificate Is Universal, Not Sampled

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=linfinity, dimension=18. Give (norm_name,dimension,qualified?,positivity+absolute_homogeneity+triangle_inequality_by_named_formula,not_sampled).

View answer and explanation

(linfinity, 18, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled)

The absolute-value identities and inequalities prove positivity, absolute homogeneity and the triangle inequality for all vectors and real scalars in this dimension. No sampled vectors establish the axioms. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 18, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-named-norm-certificate · candidate

69. A Named-Formula Certificate Is Universal, Not Sampled

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=l1, dimension=19. Give (norm_name,dimension,qualified?,positivity+absolute_homogeneity+triangle_inequality_by_named_formula,not_sampled).

View answer and explanation

(l1, 19, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled)

The absolute-value identities and inequalities prove positivity, absolute homogeneity and the triangle inequality for all vectors and real scalars in this dimension. No sampled vectors establish the axioms. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 19, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-named-norm-certificate · candidate

70. A Named-Formula Certificate Is Universal, Not Sampled

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=linfinity, dimension=20. Give (norm_name,dimension,qualified?,positivity+absolute_homogeneity+triangle_inequality_by_named_formula,not_sampled).

View answer and explanation

(linfinity, 20, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled)

The absolute-value identities and inequalities prove positivity, absolute homogeneity and the triangle inequality for all vectors and real scalars in this dimension. No sampled vectors establish the axioms. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 20, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-named-norm-certificate · candidate

71. A Named-Formula Certificate Is Universal, Not Sampled

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=l1, dimension=21. Give (norm_name,dimension,qualified?,positivity+absolute_homogeneity+triangle_inequality_by_named_formula,not_sampled).

View answer and explanation

(l1, 21, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled)

The absolute-value identities and inequalities prove positivity, absolute homogeneity and the triangle inequality for all vectors and real scalars in this dimension. No sampled vectors establish the axioms. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 21, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-named-norm-certificate · candidate

72. A Named-Formula Certificate Is Universal, Not Sampled

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=linfinity, dimension=22. Give (norm_name,dimension,qualified?,positivity+absolute_homogeneity+triangle_inequality_by_named_formula,not_sampled).

View answer and explanation

(linfinity, 22, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled)

The absolute-value identities and inequalities prove positivity, absolute homogeneity and the triangle inequality for all vectors and real scalars in this dimension. No sampled vectors establish the axioms. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 22, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-named-norm-certificate · candidate

73. A Named-Formula Certificate Is Universal, Not Sampled

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=l1, dimension=23. Give (norm_name,dimension,qualified?,positivity+absolute_homogeneity+triangle_inequality_by_named_formula,not_sampled).

View answer and explanation

(l1, 23, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled)

The absolute-value identities and inequalities prove positivity, absolute homogeneity and the triangle inequality for all vectors and real scalars in this dimension. No sampled vectors establish the axioms. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 23, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-named-norm-certificate · candidate

74. A Named-Formula Certificate Is Universal, Not Sampled

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=linfinity, dimension=24. Give (norm_name,dimension,qualified?,positivity+absolute_homogeneity+triangle_inequality_by_named_formula,not_sampled).

View answer and explanation

(linfinity, 24, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled)

The absolute-value identities and inequalities prove positivity, absolute homogeneity and the triangle inequality for all vectors and real scalars in this dimension. No sampled vectors establish the axioms. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, 24, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-named-norm-certificate · candidate

75. A Named-Formula Certificate Is Universal, Not Sampled

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=l1, dimension=25. Give (norm_name,dimension,qualified?,positivity+absolute_homogeneity+triangle_inequality_by_named_formula,not_sampled).

View answer and explanation

(l1, 25, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled)

The absolute-value identities and inequalities prove positivity, absolute homogeneity and the triangle inequality for all vectors and real scalars in this dimension. No sampled vectors establish the axioms. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, 25, Yes, positivity+absolute_homogeneity+triangle_inequality_by_named_formula, not_sampled).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-named-norm-certificate · candidate

76. The Norm of an Ordered Vector Difference

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=l1, left=[-12, -3, -5], right=[12, -6, -2] with equal dimension. Give (norm_name,left,right,left−right,distance).

View answer and explanation

(l1, [-12, -3, -5], [12, -6, -2], [-24, 3, -3], 30)

Subtract right from left coordinatewise before applying the chosen norm. Preserve the signed difference even though the distance is symmetric. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, [-12, -3, -5], [12, -6, -2], [-24, 3, -3], 30).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-induced-distance · candidate

77. The Norm of an Ordered Vector Difference

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=linfinity, left=[-11, -2, -2], right=[11, -1, -1] with equal dimension. Give (norm_name,left,right,left−right,distance).

View answer and explanation

(linfinity, [-11, -2, -2], [11, -1, -1], [-22, -1, -1], 22)

Subtract right from left coordinatewise before applying the chosen norm. Preserve the signed difference even though the distance is symmetric. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, [-11, -2, -2], [11, -1, -1], [-22, -1, -1], 22).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-induced-distance · candidate

78. The Norm of an Ordered Vector Difference

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=l1, left=[-10, -1, 1], right=[10, 4, 0] with equal dimension. Give (norm_name,left,right,left−right,distance).

View answer and explanation

(l1, [-10, -1, 1], [10, 4, 0], [-20, -5, 1], 26)

Subtract right from left coordinatewise before applying the chosen norm. Preserve the signed difference even though the distance is symmetric. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, [-10, -1, 1], [10, 4, 0], [-20, -5, 1], 26).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-induced-distance · candidate

79. The Norm of an Ordered Vector Difference

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=linfinity, left=[-9, 0, 4], right=[9, -4, 1] with equal dimension. Give (norm_name,left,right,left−right,distance).

View answer and explanation

(linfinity, [-9, 0, 4], [9, -4, 1], [-18, 4, 3], 18)

Subtract right from left coordinatewise before applying the chosen norm. Preserve the signed difference even though the distance is symmetric. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, [-9, 0, 4], [9, -4, 1], [-18, 4, 3], 18).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-induced-distance · candidate

80. The Norm of an Ordered Vector Difference

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=l1, left=[-8, 1, -4], right=[8, 1, -2] with equal dimension. Give (norm_name,left,right,left−right,distance).

View answer and explanation

(l1, [-8, 1, -4], [8, 1, -2], [-16, 0, -2], 18)

Subtract right from left coordinatewise before applying the chosen norm. Preserve the signed difference even though the distance is symmetric. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, [-8, 1, -4], [8, 1, -2], [-16, 0, -2], 18).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-induced-distance · candidate

81. The Norm of an Ordered Vector Difference

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=linfinity, left=[-7, 2, -1], right=[7, 6, -1] with equal dimension. Give (norm_name,left,right,left−right,distance).

View answer and explanation

(linfinity, [-7, 2, -1], [7, 6, -1], [-14, -4, 0], 14)

Subtract right from left coordinatewise before applying the chosen norm. Preserve the signed difference even though the distance is symmetric. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, [-7, 2, -1], [7, 6, -1], [-14, -4, 0], 14).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-induced-distance · candidate

82. The Norm of an Ordered Vector Difference

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=l1, left=[-6, 3, 2], right=[6, -2, 0] with equal dimension. Give (norm_name,left,right,left−right,distance).

View answer and explanation

(l1, [-6, 3, 2], [6, -2, 0], [-12, 5, 2], 19)

Subtract right from left coordinatewise before applying the chosen norm. Preserve the signed difference even though the distance is symmetric. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, [-6, 3, 2], [6, -2, 0], [-12, 5, 2], 19).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-induced-distance · candidate

83. The Norm of an Ordered Vector Difference

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=linfinity, left=[-5, -3, 5], right=[5, 3, 1] with equal dimension. Give (norm_name,left,right,left−right,distance).

View answer and explanation

(linfinity, [-5, -3, 5], [5, 3, 1], [-10, -6, 4], 10)

Subtract right from left coordinatewise before applying the chosen norm. Preserve the signed difference even though the distance is symmetric. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, [-5, -3, 5], [5, 3, 1], [-10, -6, 4], 10).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-induced-distance · candidate

84. The Norm of an Ordered Vector Difference

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=l1, left=[-4, -2, -3], right=[4, -5, -2] with equal dimension. Give (norm_name,left,right,left−right,distance).

View answer and explanation

(l1, [-4, -2, -3], [4, -5, -2], [-8, 3, -1], 12)

Subtract right from left coordinatewise before applying the chosen norm. Preserve the signed difference even though the distance is symmetric. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, [-4, -2, -3], [4, -5, -2], [-8, 3, -1], 12).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-induced-distance · candidate

85. The Norm of an Ordered Vector Difference

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=linfinity, left=[-3, -1, 0], right=[3, 0, -1] with equal dimension. Give (norm_name,left,right,left−right,distance).

View answer and explanation

(linfinity, [-3, -1, 0], [3, 0, -1], [-6, -1, 1], 6)

Subtract right from left coordinatewise before applying the chosen norm. Preserve the signed difference even though the distance is symmetric. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, [-3, -1, 0], [3, 0, -1], [-6, -1, 1], 6).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-induced-distance · candidate

86. The Norm of an Ordered Vector Difference

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=l1, left=[-2, 0, 3], right=[2, 5, 0] with equal dimension. Give (norm_name,left,right,left−right,distance).

View answer and explanation

(l1, [-2, 0, 3], [2, 5, 0], [-4, -5, 3], 12)

Subtract right from left coordinatewise before applying the chosen norm. Preserve the signed difference even though the distance is symmetric. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, [-2, 0, 3], [2, 5, 0], [-4, -5, 3], 12).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-induced-distance · candidate

87. The Norm of an Ordered Vector Difference

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=linfinity, left=[-1, 1, -5], right=[1, -3, 1] with equal dimension. Give (norm_name,left,right,left−right,distance).

View answer and explanation

(linfinity, [-1, 1, -5], [1, -3, 1], [-2, 4, -6], 6)

Subtract right from left coordinatewise before applying the chosen norm. Preserve the signed difference even though the distance is symmetric. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, [-1, 1, -5], [1, -3, 1], [-2, 4, -6], 6).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-induced-distance · candidate

88. The Norm of an Ordered Vector Difference

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=l1, left=[0, 2, -2], right=[0, 2, -2] with equal dimension. Give (norm_name,left,right,left−right,distance).

View answer and explanation

(l1, [0, 2, -2], [0, 2, -2], [0, 0, 0], 0)

Subtract right from left coordinatewise before applying the chosen norm. Preserve the signed difference even though the distance is symmetric. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, [0, 2, -2], [0, 2, -2], [0, 0, 0], 0).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-induced-distance · candidate

89. The Norm of an Ordered Vector Difference

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=linfinity, left=[1, 3, 1], right=[-1, -6, -1] with equal dimension. Give (norm_name,left,right,left−right,distance).

View answer and explanation

(linfinity, [1, 3, 1], [-1, -6, -1], [2, 9, 2], 9)

Subtract right from left coordinatewise before applying the chosen norm. Preserve the signed difference even though the distance is symmetric. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, [1, 3, 1], [-1, -6, -1], [2, 9, 2], 9).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-induced-distance · candidate

90. The Norm of an Ordered Vector Difference

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=l1, left=[2, -3, 4], right=[-2, -1, 0] with equal dimension. Give (norm_name,left,right,left−right,distance).

View answer and explanation

(l1, [2, -3, 4], [-2, -1, 0], [4, -2, 4], 10)

Subtract right from left coordinatewise before applying the chosen norm. Preserve the signed difference even though the distance is symmetric. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, [2, -3, 4], [-2, -1, 0], [4, -2, 4], 10).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-induced-distance · candidate

91. The Norm of an Ordered Vector Difference

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=linfinity, left=[3, -2, -4], right=[-3, 4, 1] with equal dimension. Give (norm_name,left,right,left−right,distance).

View answer and explanation

(linfinity, [3, -2, -4], [-3, 4, 1], [6, -6, -5], 6)

Subtract right from left coordinatewise before applying the chosen norm. Preserve the signed difference even though the distance is symmetric. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, [3, -2, -4], [-3, 4, 1], [6, -6, -5], 6).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-induced-distance · candidate

92. The Norm of an Ordered Vector Difference

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=l1, left=[4, -1, -1], right=[-4, -4, -2] with equal dimension. Give (norm_name,left,right,left−right,distance).

View answer and explanation

(l1, [4, -1, -1], [-4, -4, -2], [8, 3, 1], 12)

Subtract right from left coordinatewise before applying the chosen norm. Preserve the signed difference even though the distance is symmetric. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, [4, -1, -1], [-4, -4, -2], [8, 3, 1], 12).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-induced-distance · candidate

93. The Norm of an Ordered Vector Difference

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=linfinity, left=[5, 0, 2], right=[-5, 1, -1] with equal dimension. Give (norm_name,left,right,left−right,distance).

View answer and explanation

(linfinity, [5, 0, 2], [-5, 1, -1], [10, -1, 3], 10)

Subtract right from left coordinatewise before applying the chosen norm. Preserve the signed difference even though the distance is symmetric. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, [5, 0, 2], [-5, 1, -1], [10, -1, 3], 10).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-induced-distance · candidate

94. The Norm of an Ordered Vector Difference

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=l1, left=[6, 1, 5], right=[-6, 6, 0] with equal dimension. Give (norm_name,left,right,left−right,distance).

View answer and explanation

(l1, [6, 1, 5], [-6, 6, 0], [12, -5, 5], 22)

Subtract right from left coordinatewise before applying the chosen norm. Preserve the signed difference even though the distance is symmetric. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, [6, 1, 5], [-6, 6, 0], [12, -5, 5], 22).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-induced-distance · candidate

95. The Norm of an Ordered Vector Difference

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=linfinity, left=[7, 2, -3], right=[-7, -2, 1] with equal dimension. Give (norm_name,left,right,left−right,distance).

View answer and explanation

(linfinity, [7, 2, -3], [-7, -2, 1], [14, 4, -4], 14)

Subtract right from left coordinatewise before applying the chosen norm. Preserve the signed difference even though the distance is symmetric. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, [7, 2, -3], [-7, -2, 1], [14, 4, -4], 14).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-induced-distance · candidate

96. The Norm of an Ordered Vector Difference

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=l1, left=[8, 3, 0], right=[-8, 3, -2] with equal dimension. Give (norm_name,left,right,left−right,distance).

View answer and explanation

(l1, [8, 3, 0], [-8, 3, -2], [16, 0, 2], 18)

Subtract right from left coordinatewise before applying the chosen norm. Preserve the signed difference even though the distance is symmetric. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, [8, 3, 0], [-8, 3, -2], [16, 0, 2], 18).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-induced-distance · candidate

97. The Norm of an Ordered Vector Difference

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=linfinity, left=[9, -3, 3], right=[-9, -5, -1] with equal dimension. Give (norm_name,left,right,left−right,distance).

View answer and explanation

(linfinity, [9, -3, 3], [-9, -5, -1], [18, 2, 4], 18)

Subtract right from left coordinatewise before applying the chosen norm. Preserve the signed difference even though the distance is symmetric. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, [9, -3, 3], [-9, -5, -1], [18, 2, 4], 18).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-induced-distance · candidate

98. The Norm of an Ordered Vector Difference

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=l1, left=[10, -2, -5], right=[-10, 0, 0] with equal dimension. Give (norm_name,left,right,left−right,distance).

View answer and explanation

(l1, [10, -2, -5], [-10, 0, 0], [20, -2, -5], 27)

Subtract right from left coordinatewise before applying the chosen norm. Preserve the signed difference even though the distance is symmetric. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, [10, -2, -5], [-10, 0, 0], [20, -2, -5], 27).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-induced-distance · candidate

99. The Norm of an Ordered Vector Difference

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=linfinity, left=[11, -1, -2], right=[-11, 5, 1] with equal dimension. Give (norm_name,left,right,left−right,distance).

View answer and explanation

(linfinity, [11, -1, -2], [-11, 5, 1], [22, -6, -3], 22)

Subtract right from left coordinatewise before applying the chosen norm. Preserve the signed difference even though the distance is symmetric. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (linfinity, [11, -1, -2], [-11, 5, 1], [22, -6, -3], 22).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-induced-distance · candidate

100. The Norm of an Ordered Vector Difference

Only the named real-coordinate formulas l1(v)=sum_i |v_i| and linfinity(v)=max_i |v_i| are supported here. Their universal algebraic certificates, not a finite sample, supply the norm axioms. An unsupported name is outside this protocol, not a claim that every other formula fails to be a norm. norm_name=l1, left=[12, 0, 1], right=[-12, -3, -2] with equal dimension. Give (norm_name,left,right,left−right,distance).

View answer and explanation

(l1, [12, 0, 1], [-12, -3, -2], [24, 3, 3], 30)

Subtract right from left coordinatewise before applying the chosen norm. Preserve the signed difference even though the distance is symmetric. An exact finite or explicitly certified slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (l1, [12, 0, 1], [-12, -3, -2], [24, 3, 3], 30).

Reviewed step-by-step derivation

Real normed vector space · qf-mod26-norm-induced-distance · candidate

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