Let $(X , \langle \cdot , \cdot \rangle)$ be a prehilbert space and let $M \subset X$ be a vector subspace such that $(M , {\langle \cdot , \cdot \rangle}_M)$ is a Hilbert space, being $$ {\langle \cdot , \cdot \rangle}_M = {\langle \cdot , \cdot \rangle}\big|_{M \times M} : M \times M \to \mathbb{R} \mbox{ (or } \mathbb{C} \mbox{)}\mbox{.} $$ Let $P_M$ and $P_{M^{\perp}}$ denote the orthogonal projections onto $M$ and $M^\perp$, respectively. Prove that $$ \|x - u\| = \inf\{\|x - z\| : z \in M^{\perp}\}\mbox{,} $$ where $u = x - P_M(x)$ and $\|\cdot\|$ is the norm induced by $\langle \cdot , \cdot \rangle$ on $X$.
A property in Hilbert spaces
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functional-analysis
hilbert-spaces
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0$\lVert x-z\rVert^2 = \lVert x - u + u-z\rVert^2$ – 2017-02-09
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0What can I do with this expression? – 2017-02-09
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0@joseabp91: expand it and use some of your assumptions... – 2017-02-09
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0@user251257: In this case your comment doesn't give me much help. – 2017-02-09
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0Use Daniel's suggestion and think Pythagoras... – 2017-02-09