ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedaudited 2026-09-22

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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.

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Distance to a closed subspace

Example

Assume the Axiom of Countable Choice. Let $M$ be a closed linear subspace of a real or complex Hilbert space $H$ , let $x \in H$ and let $P_{M}$ be the Hilbert projection. Then for every $m \in M$

$∥ x - m ∥^{2} = ∥ x - P_{M} x ∥^{2} + ∥ P_{M} x - m ∥^{2},$

and consequently

$dist (x, M) = \inf_{m \in M} ∥ x - m ∥ = ∥ x - P_{M} x ∥,$

the infimum being attained uniquely at $m = P_{M} x$ .

Facts & Assumptions

[A1]

$P_{M} x \in M$ and $x - P_{M} x \in M^{⊥}$ , and $M$ is a linear subspace (The Hilbert orthogonal projection onto a closed subspace).

[A2]

For pairwise orthogonal vectors $∥ u + v ∥^{2} = ∥ u ∥^{2} + ∥ v ∥^{2}$ (Pythagoras and finite orthogonal sums).

[A3]

A vector of $M^{⊥}$ is orthogonal to every vector of $M$ , and $M^{⊥}$ is closed under addition (Orthogonality and the orthogonal complement).

[A4]

Countable Choice is the hypothesis under which $P_{M}$ is defined (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Verification

technique · direct

Given: Countable Choice, a closed subspace $M$ of a Hilbert space $H$ , a vector $x$ and the projection $P_{M} x$ .

1.1

For $m \in M$ write $x - m = (x - P_{M} x) + (P_{M} x - m)$ ; the first summand lies in $M^{⊥}$ and the second in $M$ , so the two are orthogonal and Pythagoras gives $∥ x - m ∥^{2} = ∥ x - P_{M} x ∥^{2} + ∥ P_{M} x - m ∥^{2}$ .

A1A2A3A4

2.1

Since $∥ P_{M} x - m ∥^{2} \geq 0$ , step 1.1 gives $∥ x - m ∥ \geq ∥ x - P_{M} x ∥$ for every $m \in M$ , with equality exactly at $m = P_{M} x$ ; hence the infimum of the distances is $∥ x - P_{M} x ∥$ , attained uniquely there.

step 1.1A1∎

Depends on

Used by

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Dependency tree · two levels

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Sources

Theo Bühler and Dietmar Salamon, Functional Analysis, §1.3.3, pp.39–41 (standard reference, not scraped)
Andrew Lin and Casey Rodriguez, MIT 18.102 Introduction to Functional Analysis, Theorem 178 (standard reference, not scraped)