TheoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedaudited 2026-09-22

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Orthogonal decomposition by a closed subspace

Statement

Assume the Axiom of Countable Choice. Let $M$ be a closed linear subspace of a real or complex Hilbert space $H$ . Then every $x \in H$ has a unique decomposition

$x = m + n, m \in M, n \in M^{⊥},$

so that $H = M \oplus M^{⊥}$ as a direct sum of the subspace $M$ and its orthogonal complement.

Facts & Assumptions

[A1]

A linear subspace is convex and contains $0$ , and the nearest point of a nonempty closed convex subset of a Hilbert space exists and is unique (Linear subspace of a vector space, Projection onto a nonempty closed convex set).

[A2]

A point $p$ is the nearest point of a closed convex set $C$ to $x$ exactly when $Re ⟨ x - p, y - p ⟩ \leq 0$ for every $y \in C$ (Variational characterisation of the nearest point).

[A3]

The pairing is linear in the first argument and conjugate-linear in the second, $S^{⊥} = {v : ⟨ v, s ⟩ = 0 \forall s \in S}$ is a linear subspace, and $v \in S^{⊥} \cap S^{⊥ ⊥}$ with $⟨ v, v ⟩ = 0$ forces $v = 0$ (Orthogonality and the orthogonal complement, Real and complex inner-product spaces and their induced length).

[A4]

Countable Choice is the selection principle consumed by the nearest-point theorem (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Proof

technique · direct

Given: Countable Choice, a Hilbert space $H$ , a closed linear subspace $M \subseteq H$ and a vector $x \in H$ .

1.1

Since $M$ is a nonempty closed convex set, $x$ has a unique nearest point $p$ in $M$ .

A1A4

2.1

The variational inequality gives $Re ⟨ x - p, w - p ⟩ \leq 0$ for every $w \in M$ . For $u \in M$ , take $w = p + u$ and $w = p - u$ to obtain $Re ⟨ x - p, u ⟩ = 0$ . Over $R$ the pairing is real-valued, so this already gives $⟨ x - p, u ⟩ = 0$ . Over $C$ , also $i u \in M$ , and the same real-part conclusion applied to $i u$ gives $0 = Re ⟨ x - p, i u ⟩ = Re (- i ⟨ x - p, u ⟩) = Im ⟨ x - p, u ⟩$ . Thus in either scalar field $⟨ x - p, u ⟩ = 0$ for every $u \in M$ , that is $x - p \in M^{⊥}$ .

step 1.1A1A2A3algebra

3.1

Setting $m = p$ and $n = x - p$ gives a decomposition $x = m + n$ with $m \in M$ and $n \in M^{⊥}$ .

step 2.1A3

4.1

If $x = m_{1} + n_{1} = m_{2} + n_{2}$ are two such decompositions, then $v = m_{1} - m_{2} = n_{2} - n_{1}$ lies in $M \cap M^{⊥}$ , since both $M$ and $M^{⊥}$ are linear subspaces, so $⟨ v, v ⟩ = 0$ and $v = 0$ ; hence $m_{1} = m_{2}$ , $n_{1} = n_{2}$ , and the decomposition is unique.

step 3.1A3algebra∎

Depends on

Used by

Orthonormal eigenbasis for a compact self adjoint operator Corollary
Deficiency subspaces and deficiency indices Definition
Fredholm determinant of a trace-class operator Definition
Isometry coisometry and partial isometry Definition
Projection valued measure Definition
The Hilbert orthogonal projection onto a closed subspace Definition
Maximal orthogonal family of cyclic reducing subspaces Lemma
Positive square root of a compact positive operator Lemma
L two kernels give Hilbert–Schmidt operators Theorem
Numerical radius is an equivalent operator norm Theorem
Partial isometry characterizations Theorem
Polar decomposition for bounded operators Theorem
Riesz representation for Hilbert spaces Theorem
Singular value decomposition for compact operators Theorem
Spectral theorem for compact self adjoint operators Theorem
The double orthogonal complement of a subspace is its closure Theorem
Toeplitz hausdorff Theorem

Dependency tree · two levels

31 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 5.37, p.238 (standard reference, not scraped)
Andrew Lin and Casey Rodriguez, MIT 18.102 Introduction to Functional Analysis, Theorem 180 (standard reference, not scraped)
Bruce Blackadar, Ilijas Farah and Asaf Karagila, Hilbert spaces without the Countable Axiom of Choice, Corollary 2.0.5 (standard reference, not scraped)