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

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Projection onto a nonempty closed convex set

Statement

Assume the Axiom of Countable Choice. Let $H$ be a real or complex Hilbert space, let $C \subseteq H$ be nonempty, closed and convex, and let $x \in H$ . Then there is exactly one point $p \in C$ with $∥ x - p ∥ = \inf_{c \in C} ∥ x - c ∥$ , that is, a unique nearest point of $C$ to $x$ .

Facts & Assumptions

[A1]

A Hilbert space is an inner-product space complete for its induced norm: every Cauchy sequence converges to a point of the space (Hilbert space).

[A2]

Every nonempty real set bounded below has an infimum, characterised by points arbitrarily close from above (Every nonempty set bounded below has an infimum, Epsilon characterisation of the infimum).

[A3]

Countable Choice selects a point from each set of a countable family of nonempty sets (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[A4]

A minimizing sequence in a nonempty convex set is Cauchy (A minimizing sequence in a convex set is Cauchy).

[A5]

$C$ is convex and $d \leq ∥ x - c ∥$ for every $c \in C$ (Convex sets and continuous real-hyperplane separation in a normed space, Greatest lower bound (infimum)). A closed set equals its closure, and a point lies in the closure whenever every ball about it meets the set (The closure of a nonempty $A$ is ${x : d (x, A) = 0}$ , equals $A$ together with its limit points, and is the smallest closed superset); limits in a metric space are unique (A sequence in a metric space has at most one limit).

[A6]

Norm distance is continuous: $∣ ∥ u ∥ - ∥ v ∥ ∣ \leq ∥ u - v ∥$ (The reverse triangle inequality in a normed space).

[A7]

Convergence of a sequence of reals to $L$ means that for every $ε > 0$ the terms are eventually within $ε$ of $L$ , and for every $ε > 0$ some $1 / n$ is below $ε$ (Limits and Cauchy sequences of reals, For every $ε > 0$ in a complete ordered field there is a natural $n \geq 1$ with $1 / n < ε$ ).

[A8]

Every inner-product norm satisfies the parallelogram law (The parallelogram law).

Proof

technique · direct

Given: Countable Choice, a real or complex Hilbert space $H$ , a nonempty closed convex set $C \subseteq H$ and a vector $x \in H$ .

1.1

The set $D = {∥ x - c ∥ : c \in C}$ is nonempty and bounded below by $0$ , so $d = \inf D$ exists and for every $n$ some $c \in C$ has $∥ x - c ∥ < d + 1 / (n + 1)$ by the epsilon characterisation of the infimum.

2.1

Countable Choice selects for every $n$ a point $c_{n} \in C$ with $∥ x - c_{n} ∥ < d + 1 / (n + 1)$ , the sets being nonempty by step 1.1.

step 1.1A3

3.1

Then $0 \leq ∥ x - c_{n} ∥ - d < 1 / (n + 1)$ for every $n$ , so $∥ x - c_{n} ∥ \to d$ by the Archimedean reciprocal bound; hence $(c_{n})$ is Cauchy by the minimizing-sequence lemma, completeness of $H$ gives a limit $p \in H$ , and closedness of $C$ places $p$ in $C$ .

step 2.1A1A4A5A7

4.1

Moreover $∥ x - p ∥ = d$ : norm continuity along the limit gives $∣ ∥ x - p ∥ - ∥ x - c_{n} ∥ ∣ \leq ∥ p - c_{n} ∥ \to 0$ , and a limit of the sequence $∥ x - c_{n} ∥$ is unique.

step 3.1A5A6

5.1

If $p, q \in C$ both satisfy $∥ x - p ∥ = ∥ x - q ∥ = d$ , then the midpoint $\frac{1}{2} (p + q)$ lies in $C$ by convexity, so $∥ x - \frac{1}{2} (p + q) ∥ \geq d$ , and the parallelogram law gives $∥ p - q ∥^{2} = 2 ∥ x - p ∥^{2} + 2 ∥ x - q ∥^{2} - 4 ∥ x - \frac{1}{2} (p + q) ∥^{2} \leq 4 d^{2} - 4 d^{2} = 0$ , whence $p = q$ : the nearest point is unique.

step 4.1A5A8algebra∎

Depends on

Used by

Orthogonal decomposition by a closed subspace Theorem
Variational characterisation of the nearest point Theorem

Dependency tree · two levels

52 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, Theorem 1.44, pp.40–41 (standard reference, not scraped)
Andrew Lin and Casey Rodriguez, MIT 18.102 Introduction to Functional Analysis, Theorem 178 (standard reference, not scraped)
Bruce Blackadar, Ilijas Farah and Asaf Karagila, Hilbert spaces without the Countable Axiom of Choice, Theorem 2.0.4 (standard reference, not scraped)