Let $V$ be a finite-dimensional complex inner product space and $U\subset V$ , a subspace of $V$ and $v \in V $, then it exists a unambiguous $w \in v + U$, such that $$\Vert w\Vert= min\{\Vert v'\Vert : v' \in v+U\}$$ While I know that this $w$ is (probably?) induced by the quotient space $V/U$ ,I have trouble understanding how to proceed.
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The problem appears to be missing some information: after $V$ is given and $U \subset V$ is given, it appears that $v \in V$ should also be given before the problem can make sense. – Lee Mosher Jun 28 '22 at 18:10
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The name "unitary space" is in disuse. – jjagmath Jun 28 '22 at 18:22
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Hint: Let $u_0$ be the orthogonal projection of $v$ to $U$ and set $w:=v-u_0$ that satisfies $w\perp U$, and use the Pythagorean theorem (if $w\perp u$ then $|w+u|^2=|w|^2+|u|^2$). – Berci Jun 28 '22 at 20:39