# complex inner product space

#### charikaar

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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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 $$\displaystyle \cos nx$$.

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