Question about inner product space inequality

**kabra** · November 24th 2009, 04:06 PM

Hi guys,

How would I show that $\sum\limits_{j = 1}^n {\left| {\left\langle {x,{v_j}} \right\rangle \left\langle {y,{v_j}} \right\rangle } \right|} \le \left\| x \right\|\left\| y \right\|$ , if $\left\{ {{v_1},{v_2},...,{v_n}} \right\}$ are an orthonormal set of vectors in V, x,y in V, V a real inner product space?

Thanks for any help!

**Shanks** · November 24th 2009, 07:37 PM

let $u=\sum_{i=1}^{n}|<x,v_i>|v_i, w=\sum_{i=1}^n|<y,v_i>|v_i$ ,
then LHS= $<u,w>\leq \left\| u \right\|\left\| w \right\|\leq \left\| x \right\|\left\| y \right\|$ =RHS

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