Norm-induced metric: proving from the norm axioms that d(x,y)=‖x−y‖ is a metric, with exact l1 and l∞ computations in ℝ³

Goal and scope

Let (V,‖·‖) be a real normed vector space. Prove that d(x,y)=‖x−y‖ is nonnegative, separates points, is symmetric and satisfies the triangle inequality; prove that the l1 and l∞ formulas on ℝⁿ are norms, so both induce metrics; then compute d1 and d∞ exactly in ℝ³ for x=(1,−2,3), y=(4,0,−1), z=(2,5,1).

Stays inside the MKO: a certified real normed vector space, and the named l1 and l∞ norms on ℝⁿ; in the general theorem both points need only lie in the same real normed space and use its norm, with no finite-dimensional hypothesis. The finite-coordinate and named-norm requirements apply only to the later ℝⁿ examples (assumption a1). The metric property is proved from the universal formula and the norm axioms; distances between finitely many points are examples only (assumption a2). Completeness and Banach spaces are not treated; the canonical MKO and its English edition are not changed.

Givens and symbols

• vector_space_axioms
• abs_rules
• order_basics
• metric_def
• finite_sum_rules

• G1 — (V,‖·‖) is a real normed vector space: (N1) ‖v‖≥0 and ‖v‖=0 ⇔ v=0; (N2) ‖tv‖=|t|‖v‖; (N3) ‖u+v‖≤‖u‖+‖v‖, for all u, v∈V and t∈ℝ.
• G2 — For x, y∈V define d(x,y)=‖x−y‖.
• G3 — For a positive integer n, on ℝⁿ: ‖v‖₁=Σ_{i=1}^{n}|v_i| and ‖v‖∞=max_{1≤i≤n}|v_i|. Example in ℝ³: x=(1,−2,3), y=(4,0,−1), z=(2,5,1).

• \|v\| — norm of the vector v
• d(x,y) — induced distance ‖x−y‖
• \|v\|_1 — l1 norm Σ|v_i|
• \|v\|_\infty — l∞ norm max|v_i|
• d_1,\ d_\infty — metrics induced by l1 and l∞
• x,\ y,\ z — three integer-coordinate points of ℝ³

Rules and sources

• vector_space_axioms · prerequisite — Vector space axioms (over ℝ or another field F): vectors form a commutative group under addition; scalar multiplication satisfies a(u+v)=au+av, (a+b)u=au+bu, (ab)u=a(bu) and 1u=u.
• abs_rules · prerequisite — Absolute value (reals): |u|=u if u≥0 and |u|=−u if u<0; |u|≥0; |uv|=|u||v|; triangle inequality |u+v|≤|u|+|v|; |u|<r ⇔ −r<u<r.
• order_basics · prerequisite — Order of the reals: exactly one of u<v, u=v, u>v holds; u<v and v<w give u<w; u<v gives u+w<v+w; u<v and w>0 give uw<vw; u<v and w<0 give uw>vw.
• metric_def · prerequisite — Metric: d:X×X→ℝ with d(x,y)≥0 and d(x,y)=0 ⇔ x=y; d(x,y)=d(y,x); triangle inequality d(x,z)≤d(x,y)+d(y,z).
• finite_sum_rules · prerequisite — Finite sums: Σ_{i=1}^{0} a_i=0 and Σ_{i=1}^{n+1} a_i=Σ_{i=1}^{n} a_i + a_{n+1}; Σ(a_i+b_i)=Σa_i+Σb_i; Σ c·a_i=c·Σa_i.
• def_unfold · prerequisite — Unfolding a definition: replacing a defined term by its definition (or folding it back) does not change whether a statement is true.
• function_def · prerequisite — Function: f:A→B assigns to each a∈A exactly one f(a)∈B. Two functions are equal ⇔ they have the same domain and codomain and agree at every a.
• eq_congruence · prerequisite — Congruence of equality: if u=v, applying the same defined operation f to both sides gives f(u)=f(v). Operating on one side only is not an instance of this rule.
• neg_mul · prerequisite — Signs and multiplication: (−u)v=−(uv) and (−u)(−v)=uv.
• mul_one · prerequisite — Multiplicative identity: 1·u=u.
• forall_intro · prerequisite — ∀-introduction (generalization): if a statement is proved for an arbitrary element x, with no extra assumption on x, it holds for every x in the domain.
• and_intro_elim · prerequisite — Conjunction: from P and Q infer P∧Q; from P∧Q infer P and infer Q.
• numeric_evaluation · prerequisite — Exact evaluation of an explicit integer or rational expression; independently recomputable.
• mko_nim_def · cited — If (V,norm) is a certified real normed vector space, d(x,y)=norm(x−y) is nonnegative, separates points, is symmetric and satisfies the triangle inequality. This slice uses exact coordinate calculations only for the named l1 and linfinity norms on R^n. · mko-norm-induced-metric
• mko_norm_def · cited — A norm sends vectors to nonnegative real numbers, with norm(v)=0 iff v=0, norm(a v)=|a| norm(v), and norm(u+v)≤norm(u)+norm(v). This slice uses only the named l1 and linfinity formulas on R^n. · mko-real-normed-vector-space
• max_rules · prerequisite — Maximum of finitely many reals: max_i w_i equals some w_k, and w_i≤max_i w_i for every i; if every w_i≤M then max_i w_i≤M.
• L_l1_norm · lemma — On ℝⁿ, ‖v‖₁=Σ|v_i| satisfies N1, N2 and N3.
• L_linf_norm · lemma — On ℝⁿ, ‖v‖∞=max|v_i| satisfies N1, N2 and N3.

Main derivation

d(x,y)=\|x-y\|

d\ge0,\ d(x,y)=0\Leftrightarrow x=y,\ d(x,y)=d(y,x),\ d(x,z)\le d(x,y)+d(y,z)

x,y\in V

x-y=x+(-1)y\in V,\ d:V\times V\to\mathbb{R}

d(x,y)=\|x-y\|

d(x,y)\ge 0

d(x,y)=0

x=y

d(x,y)=\|x-y\|

d(x,y)=\|y-x\|=d(y,x)

d(x,z)=\|x-z\|

d(x,z)\le d(x,y)+d(y,z)

\text{nonnegativity, separation, symmetry, triangle}

d(x,y)=\|x-y\|\ \text{is a metric on}\ V

\|v\|_1=\sum_{i=1}^{n}|v_i|

\|\cdot\|_1\ \text{satisfies N1, N2, N3}

\|v\|_\infty=\max_i|v_i|

\|\cdot\|_\infty\ \text{satisfies N1, N2, N3}

\|\cdot\|_1,\ \|\cdot\|_\infty\ \text{norms on}\ \mathbb{R}^n

d_1(x,y)=\|x-y\|_1,\ d_\infty(x,y)=\|x-y\|_\infty\ \text{are metrics}

x-y=(1-4,\,-2-0,\,3-(-1))=(-3,-2,4)

d_1(x,y)=3+2+4=9,\ d_\infty(x,y)=\max(3,2,4)=4

y-x=(3,2,-4)

d_1(y,x)=9=d_1(x,y)

x-z=(-1,-7,2),\ y-z=(2,-5,-2)

d_1(x,z)=10,\ d_\infty(x,z)=7,\ d_1(y,z)=9,\ d_\infty(y,z)=5

d_1(x,z)=10,\ d_\infty(x,z)=7

10\le 9+9,\ 7\le 4+5

d_1(x,y)=9,\ d_\infty(x,y)=4

d_1(x,y)\neq d_\infty(x,y)

Substeps and lemmas

x-y=0

x=y

x-y

(-1)(y-x)

\|(-1)(y-x)\|

|-1|\,\|y-x\|=\|y-x\|

x-z

(x-y)+(y-z)

\|(x-y)+(y-z)\|

\le\|x-y\|+\|y-z\|

\sum_{i}|v_i|

\sum_i|v_i|\ge0,\ \sum_i|v_i|=0\Leftrightarrow v=0

\sum_i|t\,v_i|

|t|\sum_i|v_i|

\sum_i|u_i+v_i|

\le\sum_i(|u_i|+|v_i|)=\sum_i|u_i|+\sum_i|v_i|

\max_i|v_i|

\max_i|v_i|\ge0,\ \max_i|v_i|=0\Leftrightarrow v=0

\max_i|t\,v_i|

|t|\max_i|v_i|

|u_i+v_i|\le|u_i|+|v_i|\le\|u\|_\infty+\|v\|_\infty

\|u+v\|_\infty\le\|u\|_\infty+\|v\|_\infty

Reverse audits

• S15 · premise_audit · S11, numeric_evaluation — It only compares two exact numbers, showing that the metric depends on the chosen norm.
• S14 · premise_audit · S06, order_basics — A check at finitely many points supports only those points; the general conclusion comes from S06.
• S13 · premise_audit · G3, numeric_evaluation — Exact integer computation, recomputable (N4 to N7).
• S12 · premise_audit · S05, numeric_evaluation — The general conclusion comes from S05; this step is only an example.
• S11 · premise_audit · G3, numeric_evaluation — Exact integer computation, recomputable (N1, N2).
• S10 · premise_audit · S07, S08, S09 — S07 needs the norm axioms, which S08 and S09 supply; each metric is paired only with the norm that induces it.
• S09.3 · premise_audit · abs_rules, max_rules, order_basics — The upper-bound argument gives a one-way inequality; this sub-step does not use L_linf_norm.
• S09.2 · inverse_equivalence · max_rules, abs_rules, order_basics — Both sides are the same number; the condition is |t|≥0 (multiplying by a negative number would reverse inequalities). This sub-step does not use L_linf_norm.
• S09.1 · premise_audit · max_rules, abs_rules — It uses that the maximum is an upper bound and is attained; this sub-step does not use L_linf_norm.
• S09 · premise_audit · G3, L_linf_norm — It needs only the l∞ formula and the lemma; it holds for every vector.
• S08.3 · premise_audit · abs_rules, order_basics — Adding inequalities term by term preserves their direction, and no converse is claimed; this sub-step does not use L_l1_norm.
• S08.2 · inverse_equivalence · abs_rules, finite_sum_rules — Both equalities can be reversed; this sub-step does not use L_l1_norm.
• S08.1 · premise_audit · abs_rules, order_basics — The key is that a zero sum of nonnegative terms forces every term to be zero; this sub-step does not use L_l1_norm.
• S08 · premise_audit · G3, L_l1_norm — It needs only the l1 formula and the lemma; it holds for every n and every vector.
• S07 · sufficient_goal_reduction · S03, S04, S05, S06, metric_def, forall_intro — Goal G: d is a metric. Sufficient condition H: the four properties hold for all points; H⇒G by the definition of a metric. H is proved forwards in S03 to S06.
• S06.2 · premise_audit · mko_norm_def, S06.1 — N3 is a one-way inequality; equality does not hold in general and is not claimed.
• S06.1 · inverse_equivalence · vector_space_axioms — Two expressions for the same vector, so the step is two-way.
• S06 · premise_audit · mko_norm_def, vector_space_axioms — It needs the vector identity (sub-step S06.1) and N3 (sub-step S06.2); this is an inequality, and no converse is claimed.
• S05.2 · inverse_equivalence · mko_norm_def, abs_rules — N2 is an equation and |−1|=1, so both sides are the same number.
• S05.1 · inverse_equivalence · vector_space_axioms — Two expressions for the same vector, so the step is two-way.
• S05 · inverse_equivalence · mko_norm_def, vector_space_axioms, abs_rules — Both sides are the same real number (sub-steps S05.1 and S05.2 give this from the vector space axioms and N2); swapping x and y and repeating the argument goes back.
• S04.1 · inverse_equivalence · vector_space_axioms — Adding y and subtracting y undo each other; both directions come from the additive group axioms.
• S04 · inverse_equivalence · mko_norm_def, vector_space_axioms — The second half of N1 is itself an iff, and so is x−y=0⇔x=y, proved in sub-step S04.1, so the whole chain holds both ways.
• S03 · premise_audit · G1, mko_norm_def — It uses only the first half of N1 and holds for every x, y.
• S02 · premise_audit · vector_space_axioms, G2 — It only confirms that d is defined; if the two points were in different spaces, x−y would be meaningless.
• S01 · premise_audit · mko_nim_def, metric_def — This step only lists the obligations and derives no new fact.

Error controls and negative controls

A finite sample does not establish the metric or its origin

The common mistake named by the MKO is to claim, after checking finitely many distances, that a metric is induced by a norm. The 10≤18 and 7≤9 of S14 are only examples. Counterexample: the discrete metric on ℝ³ (distance 1 between distinct points) satisfies the metric axioms, but a norm-induced metric always has d(2v,0)=‖2v‖=2d(v,0), while the discrete metric gives 1≠2 for v≠0, so no norm induces it.

N1 and N3 alone do not ensure symmetry

N1 and N3 alone are not enough. On ℝ, f(v)=max(v,−2v) is nonnegative, zero only at 0, and subadditive (a maximum of linear functions), but f(1)=1 and f(−1)=2, so f(x−y) is not symmetric: from 1 to 0 it is 1, from 0 to 1 it is 2. S05.2 obtains symmetry exactly through N2 with t=−1.

One named norm, one dimension

d1(x,y)=9 while d∞(x,y)=4 (S15). Mixing norms requires a comparison estimate in addition to the triangle inequality for one metric. In this example d∞(x,y)≤d1(x,y), so d∞(x,z)≤d∞(x,y)+d∞(y,z)≤d1(x,y)+d∞(y,z) does hold. When the two points have different dimensions, x−y is undefined in this model (S02).

• d(x,y)=‖x‖₁−‖y‖₁ is a metric on ℝ³ → reject — x=0 and y=(1,0,0) give 0−1=−1<0, violating nonnegativity.
• The discrete metric on ℝ³ is induced by some norm → reject — An induced metric satisfies d(2v,0)=2d(v,0), but the discrete metric gives 1≠2 for v≠0.
• The triangle inequality was checked at x, y, z (10≤18, 7≤9), so d1 and d∞ are metrics → reject — A finite sample supports only those points; the metric property comes from the universal proof in S06 to S10.
• The l1 distance between (1,−2,3)∈ℝ³ and (1,2)∈ℝ² is 4 → reject — The points are not in the same space, so x−y is undefined (assumption a1).

Result and limitations

For every real normed vector space, d(x,y)=‖x−y‖ is nonnegative (N1), separates points (N1 with x−y=0⇔x=y), is symmetric (N2 with t=−1) and satisfies the triangle inequality (x−z=(x−y)+(y−z) and N3), so it is a metric. The l1 and l∞ formulas on ℝⁿ are norms, so d1 and d∞ are metrics. Example: d1(x,y)=9, d∞(x,y)=4, d1(x,z)=10, d∞(x,z)=7, d1(y,z)=9, d∞(y,z)=5.

• max_rules is a premise of this guide and is not derived here from the order axioms of the reals.
• The subadditivity of f(v)=max(v,−2v) in the error controls is explanatory (a standard property of a maximum of linear functions) and is not a numbered step.

Verification and source binding

12 declared checks for this guide; only the listed expressions are covered.

• source commit: b28a5edbd46279b1db2a3f3f48bb3f447e18899f
• adopted source SHA-256: 49854f1c5f3eec27f6de9fba35448a82d0db140a86fbabd4f87f370e0e39ac1a
• original source SHA-256: 6d24d167b92a2af8ec2a7ab0a267082c9664a86cde10e8728624bf97a61aff02
• Lean: NOT_RUN · independent AI: NOT_MEASURED

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