TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-16

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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.
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Every linear map between finite-dimensional inner product spaces has a unique adjoint

Statement

Every linear map $T : V \to W$ between finite-dimensional real or complex inner product spaces has a unique adjoint $T^{*} : W \to V$ .

Facts & Assumptions

Given: A linear map $T : V \to W$ between finite-dimensional inner product spaces.

[L1]

Every linear functional on a finite-dimensional inner product space has a unique representing vector (Finite-dimensional Riesz representation: every functional is uniquely $v \mapsto ⟨ v, w ⟩$ ).

[L2]

An adjoint must satisfy $⟨ T v, w ⟩_{W} = ⟨ v, T^{*} w ⟩_{V}$ for all $v, w$ (The adjoint $T^{*} : W \to V$ is characterised by $⟨ T v, w ⟩_{W} = ⟨ v, T^{*} w ⟩_{V}$ ).

[L3]

Inner products separate vectors: equality of all pairings forces equality of the paired vectors (Inner products separate vectors, and the induced norm is homogeneous: $∥ λ v ∥ = ∣ λ ∣ ∥ v ∥$ ).

Proof

technique · direct

1.1L1L2

Fix $w \in W$ . The function $v \mapsto ⟨ T v, w ⟩_{W}$ is a linear functional on $V$ , so [L1] supplies a unique vector, call it $T^{*} w$ , satisfying [L2].

2.1step 1.1L3algebra

For scalars $a, b$ and $w_{1}, w_{2} \in W$ , pairing the representatives from step 1.1 shows $T^{*} (a w_{1} + b w_{2})$ and $a T^{*} w_{1} + b T^{*} w_{2}$ have the same pairing with every $v$ . By [L3] they are equal. Thus $T^{*}$ is linear.

3.1step 1.1step 2.1L1∎

Any adjoint must assign to each $w$ the unique representative from step 1.1, so it equals $T^{*}$ . This proves existence and uniqueness, including when either space is zero.

Depends on

Used by

Dependency tree · two levels

11 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Sheldon Axler, Linear Algebra Done Right, 4th ed., discussion following definition 7.1 (standard reference, not scraped)
Sergei Treil, Linear Algebra Done Wrong, §5.5.1 (standard reference, not scraped)