DefinitionDefinition: Literature-sourcedProof: Not applicablejudge 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.

The adjoint $T^{} : W \to V$ is characterised by $⟨ T v, w ⟩_{W} = ⟨ v, T^{} w ⟩_{V}$

Definition

Let $V, W$ be inner product spaces (Real and complex inner product spaces, with the inner product linear in the first argument) over the same field and let $T : V \to W$ be linear (Linear map between vector spaces over the same field). An adjoint of $T$ is a linear map $T^{*} : W \to V$ such that

$⟨ T v, w ⟩_{W} = ⟨ v, T^{*} w ⟩_{V}$

for every $v \in V$ and $w \in W$ . Existence is not part of the definition; in finite dimensions it will follow from Riesz representation.

Definition references

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

Linear map between vector spaces over the same field

The adjoint $T^{} : W \to V$ is characterised by $⟨ T v, w ⟩_{W} = ⟨ v, T^{} w ⟩_{V}$

Definition

Definition references

Depends on

Dependency tree · 2 levels

Sources

The adjoint T∗:W→VT∗:W→V is characterised by ⟨Tv,w⟩W=⟨v,T∗w⟩V⟨Tv,w⟩W​=⟨v,T∗w⟩V​

Definition

Definition references

Depends on

Dependency tree · 2 levels

Sources

The adjoint $T^{} : W \to V$ is characterised by $⟨ T v, w ⟩_{W} = ⟨ v, T^{} w ⟩_{V}$