A finite-dimensional Hilbert certificate from one matrix

Goal and scope

Verify the consistent inner product, norm, metric and completeness of R^2 with A=[[2,1],[1,2]].

Only this finite-dimensional real carrier; no infinite-dimensional, compactness or separability claim.

Givens and symbols

• algebra
• inner_norm_metric
• finite_dim_complete

• G1 — V=R^2, A=[[2,1],[1,2]]; define <x,y>_A=x^T A y and derive both norm and metric from this same inner product.

• A — fixed positive-definite matrix
• \|\ \|_A,d_A — matching norm and metric

Rules and sources

• mko_definition · cited — The carrier, inner product, induced norm, induced metric and completeness certificate must describe the same space. This slice uses named completeness-theorem certificates for finite-dimensional R^n with a standard or positive-definite-matrix inner product. Sampled Cauchy sequences do not establish completeness. · mko-real-hilbert-space
• algebra · prerequisite — Use real-field identities; division requires a nonzero denominator and multiplication by a positive number preserves inequalities.
• inner_norm_metric · prerequisite — A positive-definite symmetric bilinear form defines a real inner product; its square-root self-product is a norm, including the triangle inequality, and the difference norm is a metric.
• finite_dim_complete · prerequisite — Named completeness certificate: every finite-dimensional real inner-product space is complete in the norm and metric induced by that inner product.

Main derivation

x=(x_1,x_2),\ y=(y_1,y_2)

\langle x,y\rangle_A=2x_1y_1+x_1y_2+x_2y_1+2x_2y_2

\langle ax+bz,y\rangle_A

a\langle x,y\rangle_A+b\langle z,y\rangle_A

\langle x,x\rangle_A=2x_1^2+2x_1x_2+2x_2^2

\langle x,x\rangle_A=(x_1+x_2)^2+x_1^2+x_2^2\ge0

V=\mathbb R^2,\ \langle\ ,\ \rangle_A\text{ symmetric bilinear positive definite}

(V,\langle\ ,\ \rangle_A)\text{ is a real inner-product space}

\|x\|_A=\sqrt{\langle x,x\rangle_A}

\|x\|_A=\sqrt{2x_1^2+2x_1x_2+2x_2^2}

d_A(x,y)=\|x-y\|_A

d_A(x,y)=\sqrt{2(x_1-y_1)^2+2(x_1-y_1)(x_2-y_2)+2(x_2-y_2)^2}

(x_1+x_2)^2\le2(x_1^2+x_2^2)

\|x\|_2^2\le\|x\|_A^2\le3\|x\|_2^2

\dim_{\mathbb R}V=2<\infty

(V,\langle\ ,\ \rangle_A,\|\ \|_A,d_A)\text{ is a real Hilbert space}

Substeps and lemmas

No substeps in this guide.

Reverse audits

• S08 · premise_audit · G1, S04, S05, S06 — The certificate binds the same carrier, product, norm and metric; sampled Cauchy sequences cannot replace it.
• S07 · premise_audit · S03, S05 — The bounds hold for all real vectors but imply no unprovided infinite-dimensional result.
• S06 · premise_audit · S05 — Every difference vector stays in the same V.
• S05 · premise_audit · S04 — No different matrix or unbound norm may be substituted.
• S04 · premise_audit · G1, S02, S03 — The carrier remains R^2, not the subset of rational vectors.
• S03 · inverse_equivalence · S01 — No determinant shortcut or numerical sample replaces the all-x proof.
• S02 · inverse_equivalence · S01 — a,b are arbitrary real scalars, not sampled integers.
• S01 · inverse_equivalence · G1 — Swapping x,y preserves the expression, including both cross coefficients.

Error controls and negative controls

Five-way binding

Keep V, A, inner product, norm and metric consistently bound.

Named completeness

Explicitly apply finite-dimensional real-inner-product completeness, not samples.

• Replacing R^2 by Q^2 permits the same real finite-dimensional theorem. → reject — Q^2 is not a real vector space, so the carrier premise fails.
• Finitely checking Cauchy sequences proves completeness. → reject — Completeness quantifies over every Cauchy sequence.
• Hilbert therefore makes this space compact. → reject — No compactness certificate is supplied; R^2 itself is unbounded.

Result and limitations

This R^2, with all structures induced by the same A, is a real Hilbert space; completeness is supplied by the finite-dimensional completeness theorem.

Verification and source binding

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

• source commit: b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f
• adopted source SHA-256: 4930257bba12534036e760dca2b4c1277cdaeb84ac961481c36f384db87a6209
• original source SHA-256: 1dfb783bf683beff32490fed97cb7f8755ddeca7526828da95db41018e1e36fd
• Lean: NOT_RUN · independent AI: SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW

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