TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-16

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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

1.1

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].

L1L2

2.1

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.

step 1.1L3algebra

3.1

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.

step 1.1step 2.1L1∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 42 results over 13 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click 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)