Euclidean space/Distance to subspace/Orthogonal projection/Fact/Proof

Proof

For $u \in U$ , we have, due to the Pythagorean theorem,

d (v, u)^{2} = d (v, p_{U} (v))^{2} + d (p_{U} (v), u)^{2},

as $p_{U} (v) - u \in U$ and $v - p_{U} (v) \in U^{⊥}$ are perpendicular to each other. This expression is minimal if and only if $d (p_{U} (v), u) = 0$ , and this holds if and only if

p_{U} (v) = u .

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