LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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.

Inner products separate vectors, and the induced norm is homogeneous: $∥ λ v ∥ = ∣ λ ∣ ∥ v ∥$

Statement

In a real or complex inner product space:

if $⟨ u, v ⟩ = 0$ for every $v$ , then $u = 0$ ;
$∥ λ v ∥ = ∣ λ ∣ ∥ v ∥$ for every scalar $λ$ and vector $v$ .

Facts & Assumptions

Given: Vectors $u, v$ in an inner product space and a scalar $λ$ .

[L1]

Positive definiteness says $⟨ w, w ⟩ = 0$ exactly when $w = 0$ , and the inner product is linear first and conjugate-linear second (Real and complex inner product spaces, with the inner product linear in the first argument).

[L2]

The induced norm is the unique nonnegative square root of $⟨ w, w ⟩$ (The norm $∥ v ∥ = \sqrt{⟨ v, v ⟩}$ induced by a real or complex inner product, Existence and uniqueness of $n$ -th roots: a unique $a^{1 / n} \geq 0$ with $(a^{1 / n})^{n} = a$ ).

[L3]

For a complex scalar, $λ \overline{λ} = ∣ λ ∣^{2}$ (Conjugation is an involutive real-field automorphism, $z \overline{z} = ∣ z ∣^{2}$ , and modulus is definite, multiplicative, and subadditive).

Proof

technique · direct

1.1L1

If $⟨ u, v ⟩ = 0$ for every $v$ , take $v = u$ ; [L1] gives $u = 0$ .

2.1L1L2L3∎

Sesquilinearity and [L3] give $⟨ λ v, λ v ⟩ = λ \overline{λ} ⟨ v, v ⟩ = ∣ λ ∣^{2} ∥ v ∥^{2}$ . Both $∥ λ v ∥$ and $∣ λ ∣ ∥ v ∥$ are nonnegative, so uniqueness in [L2] gives their equality.

Depends on

Dependency tree · 2 levels

Sources

Sheldon Axler, Linear Algebra Done Right, 4th ed., §6A

Inner products separate vectors, and the induced norm is homogeneous: ∥λv∥=∣λ∣∥v∥∥λv∥=∣λ∣∥v∥