Originally Posted by

**AlexP** I'm having trouble with this problem. I've tried rewriting things in about every way I know how to but I haven't arrived at an answer. I'd appreciate some help.

"Let $\displaystyle V$ be an inner product space and let $\displaystyle S=\{v_1, ..., v_n\}$ be an orthonormal subset of $\displaystyle V$. Prove that for any $\displaystyle x \in V$ we have $\displaystyle ||x||^2 \ge \displaystyle\sum^n_{i=1} |\langle x,v_i \rangle |^2.$"

As a hint the book says to use the fact that $\displaystyle x \in V$ can be written uniquely as $\displaystyle w+w'$ where $\displaystyle w \in W=\mbox{span}(S)$ and $\displaystyle w' \in W^\perp$, the orthogonal complement of W, and use the fact that for $\displaystyle x$,$\displaystyle y$ orthogonal, $\displaystyle ||x+y||^2 = ||x||^2 + ||y||^2$.

Thanks for any help.