LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-22

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Kernel–range orthogonality for Hilbert adjoints

Statement

Assume the Axiom of Countable Choice. Let $T \in B (H, K)$ be a bounded linear operator between real or complex Hilbert spaces, with kernel $\ker T = {x : T x = 0}$ , range $ran T = T [H]$ and Hilbert adjoint $T^{*} \in B (K, H)$ . Then

$(ran T)^{⊥} = \ker T^{*}, (ran T^{*})^{⊥} = \ker T,$

and consequently

$\overline{ran T} = (\ker T^{*})^{⊥}, \overline{ran T^{*}} = (\ker T)^{⊥} .$

Facts & Assumptions

[A1]

$⟨ T x, y ⟩_{K} = ⟨ x, T^{*} y ⟩_{H}$ for all $x \in H$ , $y \in K$ , and the adjoint is a bounded linear operator (The Hilbert-space adjoint of a bounded operator, A bounded linear operator between normed spaces).

[A2]

$T^{* *} = T$ (Hilbert-adjoint identities).

[A3]

$S^{⊥} = {v : ⟨ v, s ⟩ = 0 \forall s \in S}$ ; if $⟨ v, w ⟩ = 0$ for every $w$ then $v = 0$ , and $S^{⊥ ⊥} = \overline{S}$ for every linear subspace $S$ (Orthogonality and the orthogonal complement, Real and complex inner-product spaces and their induced length, The double orthogonal complement of a subspace is its closure).

[A4]

The image of a linear map is a linear subspace, and the kernel of a linear map is a linear subspace (Linear subspace of a vector space).

[A5]

Countable Choice is the hypothesis under which adjoints and double complements are available (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Proof

technique · direct

Given: Countable Choice, Hilbert spaces $H, K$ and a bounded linear operator $T : H \to K$ .

1.1

For $y \in K$ , one has $y \in (ran T)^{⊥}$ exactly when $⟨ T x, y ⟩_{K} = 0$ for every $x$ , which by the adjoint identity is exactly when $⟨ x, T^{*} y ⟩_{H} = 0$ for every $x$ ; taking $x = T^{*} y$ shows this happens exactly when $T^{*} y = 0$ , that is $y \in \ker T^{*}$ .

A1A3A5

2.1

Replacing $T$ by $T^{*}$ in step 1.1 and using $T^{* *} = T$ gives $(ran T^{*})^{⊥} = \ker T$ .

step 1.1A2

3.1

The range of a linear map is a linear subspace, so the double-complement theorem applies to it: $\overline{ran T} = (ran T)^{⊥ ⊥} = ((ran T)^{⊥})^{⊥} = (\ker T^{*})^{⊥}$ , and likewise $\overline{ran T^{*}} = (\ker T)^{⊥}$ .

step 1.1step 2.1A3A4∎

Depends on

Used by

Spectrum of a positive operator is nonnegative Lemma
Spectrum of a self adjoint operator is real Lemma
Cayley correspondence between self-adjoint operators and unitaries Theorem
Numerical radius is an equivalent operator norm Theorem
Partial isometry characterizations Theorem
Polar decomposition for bounded operators Theorem

Dependency tree · two levels

33 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

Theo Bühler and Dietmar Salamon, Functional Analysis, Lemma 5.38, p.238 (standard reference, not scraped)
Andrew Lin and Casey Rodriguez, MIT 18.102 Introduction to Functional Analysis, Theorem 185 (standard reference, not scraped)