Projection onto a nonempty closed convex set

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).

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).

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

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

$C$ is convex and $d\le \mathrm{\parallel }x-c\mathrm{\parallel }$ 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).

Norm distance is continuous: $\mid \mathrm{\parallel }u\mathrm{\parallel }-\mathrm{\parallel }v\mathrm{\parallel }\mid \le \mathrm{\parallel }u-v\mathrm{\parallel }$ (The reverse triangle inequality in a normed space).

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

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