complex inner product space

Feb 2008
184
1
Let C[0,2pi] be the inner product space of complex continuous functions on [0,2pi].

let {\(\displaystyle v_k\):k=1,2,3,....} be an orthonormal sequence in C[0,2pi], given by \(\displaystyle v_k(x)=sinkx/pi\).

Give an example of a function u such \(\displaystyle u\neq\sum_{k=1}<u,v_k>v_k\) sum from k=1 to infinity.

thanks for any help.
 

Opalg

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Aug 2007
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Let C[0,2pi] be the inner product space of complex continuous functions on [0,2pi].

let {\(\displaystyle v_k\):k=1,2,3,....} be an orthonormal sequence in C[0,2pi], given by \(\displaystyle v_k(x)=sinkx/pi\).

Give an example of a function u such \(\displaystyle u\neq\sum_{k=1}<u,v_k>v_k\) sum from k=1 to infinity.
You could try the constant function 1, or a function of the form \(\displaystyle \cos nx\).