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**arbolis** I'm unable to do the following problem : Let $\displaystyle V$ be a vector space over $\displaystyle \mathbb{K}$ whose dimension is $\displaystyle n$ and with an inner product $\displaystyle <,>$. Let $\displaystyle W$ be a subspace of $\displaystyle V$ with dimension $\displaystyle m$ and $\displaystyle \{ w_1,...,w_m \}$ be an orthonormal basis of $\displaystyle W$.

We define the orthogonal projection over $\displaystyle W$ as $\displaystyle P_W(v)=\sum_{i=1}^{m} <v,w_i>w_i$.

Demonstrate that $\displaystyle ||v-P_W(v) || \leq || v-w||$, $\displaystyle \forall w \in W$.

My attempt : I suppose it's true and want to see that it's really true ( don't want to fall over any contradiction).

So $\displaystyle ||v-P_W(v) || \leq || v-w||$ $\displaystyle \Leftrightarrow ||v-P_W(v) ||^2 \leq || v-w||^2$ $\displaystyle \Leftrightarrow <v,-P_W(v)> \leq <v,-w>$. And I didn't reach anything.