(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Let V be a real inner product space, and let v_{1}, v_{2}, ... , v_{k}be a set of orthonormal vectors.

Prove

Ʃ (from j=1 to k)|<x,v_{j}><y,v_{j}>| ≤ ||x|| ||y||

When is there equality?

2. Relevant equations

3. The attempt at a solution

I've tried using the two inequalities given to us in lectures, namely Cauchy-Schwarz Inequality which states

|<v,w>| ≤ ||v|| ||w||

But surely, using this inequality, we get Ʃ (from j=1 to k)|<x,v_{j}><y,v_{j}>| ≤ k(||x|| ||v|| ||y|| ||v|| = k( ||x|| ||y||) since the v are orthonormal!

I understand this is an inequality, and so obviously the inequality above is a better approximation than the one I've just shown, but I'm not sure where to go.

The other inequality is Bessel's Inequality which states

||v||^{2}≥ Ʃ|<v, e_{i}>|^{2}if e_{i}is a set of orthonormal elements.

Thanks

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# Homework Help: Inner Product Sum Inequality

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