LemmaStatement: AI-generatedProof: AI-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Coordinate vertices inside a relative open ball

Statement

Let $E$ be a finite-dimensional coordinate Euclidean space, $T \subseteq E$ , and $u \in {int}_{E} (T)$ . If $d = \dim E > 0$ , there exist an orthonormal basis $e_{1}, \dots, e_{d}$ and $ε > 0$ such that all $p_{i}^{\pm} = u \pm ε e_{i}$ belong to $T$ , and $B_{E} (u, ε) \subseteq T$ .

Statement references

Annihilators, relative hyperplanes and complexification

Real and complex inner product spaces, with the inner product linear in the first argument

Orthonormal families, complete orthonormal systems and Hilbert bases

Every finite-dimensional real or complex inner product space has an orthonormal basis

Facts & Assumptions

[F1]

Relative interior is computed using Euclidean balls in $E$ : $u \in {int}_{E} (T)$ means that some $R > 0$ satisfies $B_{E} (u, R) \subseteq T$ (Annihilators, relative hyperplanes and complexification).

[F2]

The coordinate Euclidean inner product is $⟨ x, y ⟩ = \sum_{k = 1}^{d} x_{k} y_{k}$ ; it is a real inner product, with the bilinearity and positive definiteness of Real and complex inner product spaces, with the inner product linear in the first argument.

[F3]

An orthonormal family satisfies $⟨ e_{i}, e_{j} ⟩ = δ_{i j}$ . Its vectors have induced length $∥ e_{i} ∥ = \sqrt{⟨ e_{i}, e_{i} ⟩} = 1$ (Orthonormal families, complete orthonormal systems and Hilbert bases).

[F4]

Every finite-dimensional real inner product space has an orthonormal basis (Every finite-dimensional real or complex inner product space has an orthonormal basis).

Proof

Given: $E, T, u$ as in the statement and $d = \dim E > 0$ . Use the coordinate Euclidean inner product, the induced length $∥ x ∥ = \sqrt{⟨ x, x ⟩}$ and the distance $d_{E} (x, y) = ∥ x - y ∥$ .

Goal: Choose the opposite vertices and the ball required in the statement.

Proof technique: direct.

1.1givenF1F2F4choose

By [F1], choose $R > 0$ with $B_{E} (u, R) \subseteq T$ , and set $ε = R / 2$ . The space $E$ is a finite-dimensional real inner product space by [F2], so [F4] supplies an orthonormal basis $e_{1}, \dots, e_{d}$ .

2.1step 1.1F1F2F3algebra∎

For each $i$ , real bilinearity and orthonormality give $⟨ \pm ε e_{i}, \pm ε e_{i} ⟩ = ε^{2} ⟨ e_{i}, e_{i} ⟩ = ε^{2}$ . Hence $d_{E} (u \pm ε e_{i}, u) = ε < R$ , so $p_{i}^{\pm} = u \pm ε e_{i} \in T$ . Also $B_{E} (u, ε) \subseteq B_{E} (u, R) \subseteq T$ since $0 < ε < R$ . These are both required assertions.

Depends on

Used by · 3

Dependency tree · 2 levels

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