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Thread: Question about inner product space inequality

  1. #1
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    Question about inner product space inequality

    Hi guys,

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

    Thanks for any help!
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  2. #2
    Senior Member Shanks's Avatar
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    let$\displaystyle u=\sum_{i=1}^{n}|<x,v_i>|v_i, w=\sum_{i=1}^n|<y,v_i>|v_i$,
    then LHS=$\displaystyle <u,w>\leq \left\| u \right\|\left\| w \right\|\leq \left\| x \right\|\left\| y \right\|$=RHS
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