Let $V$ be an inner product space of finite dimension. Let $v_1, v_2, ... , v_m$ be orthogonal vectors in $V$.
How can I show that:
a. $$||v_1+v_2+...+v_m||^2=||v_1||^2+||v_2||^2+...+||v_m||^2$$
b. Let $v_1, v_2, ... , v_m$ be orthonormal vectors in $V$. let $v$ be some vector and $\alpha _i=<v,v_i>$ it's Fourier coefficient. Let $\beta _1, \beta _2, ... , \beta _n$ some scalars. Use the previous result or any other to prove that: $$||v-\sum \alpha _i v_i||\le ||v-\sum \beta _i v_i||$$
For part a.):
We have, $$ \big\langle (v_1 + v_2 + \cdots + v_m), (v_1 + v_2 + \cdots + v_m) \big\rangle = \sum_{i,j \in \{1,\dots,m\}} \langle v_i, v_j\rangle. $$ Now, since $v_1, \cdots, v_m$ are (pairwise) orthogonal, we have $\langle v_i, v_j \rangle = 0$ when $i \neq j$. Thus noting the LHS is just the norm squared of $v_1 + \cdots + v_m$, $$ ||v_1 + v_2 + \cdots + v_m||^2 = \langle v_1, v_1 \rangle + \cdots \langle v_m,v_m\rangle = ||v_1||^2 + \cdots + ||v_m||^2, $$ as desired.
Hint for part b.): think of $\alpha_i$ as the projection of $v$ onto its $i$th coordinate.