LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6.1-sol)

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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The projection onto a closed convex set is characterised by a variational inequality

Statement

Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $H$ be a real Hilbert space (Hilbert space), let $K \subseteq H$ be nonempty, closed and convex (Convex sets and continuous real-hyperplane separation in a normed space), let $x \in H$ , and let $P_{K} x$ denote the unique nearest point of $K$ to $x$ , whose existence and uniqueness are supplied by Projection onto a nonempty closed convex set. Then for every $u \in K$ one has $u = P_{K} x ⟺ ⟨ u - x, v - u ⟩ \geq 0 for every v \in K .$

Facts & Assumptions

Given: A real Hilbert space $H$ , a nonempty closed convex set $K \subseteq H$ , a vector $x \in H$ and a point $u \in K$ ; $P_{K} x$ is the unique nearest point of $K$ to $x$ .

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): Countable Choice is the selection principle consumed by the existence-and-uniqueness theorem for nearest points.

[F1]

Projection onto a nonempty closed convex set: under Countable Choice, a nonempty closed convex subset $K$ of a real or complex Hilbert space has exactly one nearest point to $x$ ; hence $P_{K} x$ is well defined and $u = P_{K} x$ holds exactly when $∥ x - u ∥ \leq ∥ x - v ∥$ for every $v \in K$ .

[F2]

Variational characterisation of the nearest point: for $p \in K$ , $p$ is the nearest point of $K$ to $x$ if and only if $Re ⟨ x - p, v - p ⟩ \leq 0$ for every $v \in K$ .

[F3]

Real and complex inner-product spaces and their induced length: the inner product of a real inner product space is real-valued, so $Re ⟨ a, b ⟩ = ⟨ a, b ⟩ = ⟨ b, a ⟩$ and $⟨ a, b ⟩ = - ⟨ - a, b ⟩$ for all vectors $a, b$ ; in particular $u - x = - (x - u)$ .

Proof

technique · direct

Given: A real Hilbert space $H$ , a nonempty closed convex set $K \subseteq H$ , vectors $x \in H$ and $u \in K$ , and the unique nearest point $P_{K} x$ of $K$ to $x$ .

1.1A1F1

By [F1] the point $u$ equals $P_{K} x$ if and only if $u$ is a nearest point of $K$ to $x$ , that is, $∥ x - u ∥ \leq ∥ x - v ∥$ for every $v \in K$ ; here Countable Choice enters through the existence and uniqueness of the nearest point recorded in [A1].

1.2F2

By [F2] the point $u$ is a nearest point of $K$ to $x$ if and only if $Re ⟨ x - u, v - u ⟩ \leq 0$ for every $v \in K$ .

2.1step 1.1step 1.2F3algebra∎

Since the inner product is real-valued [F3], $Re ⟨ x - u, v - u ⟩ = ⟨ x - u, v - u ⟩ = - ⟨ u - x, v - u ⟩$ , so the inequality of step 1.2 is equivalent to $⟨ u - x, v - u ⟩ \geq 0$ for every $v \in K$ ; combining this with step 1.1 gives $u = P_{K} x$ if and only if $⟨ u - x, v - u ⟩ \geq 0$ for every $v \in K$ , which is the asserted characterisation.

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Sources

Nguyen Dong Yen and Bui Trong Kim, Linear operators satisfying the assumptions of some generalized Lax-Milgram theorems, Acta Mathematica Vietnamica 26(3) (2001), 407-417 (standard reference, not scraped)
J. T. Oden and N. Kikuchi, Theory of variational inequalities with applications to problems of flow through porous media, International Journal of Engineering Science 18 (1980), 1173-1284 (standard reference, not scraped)