I know that for a finite dimensional inner product space,the best approximation of a vector exists in a given subspace not containing the vector.But is it true for infinite dimensional inner product space.In general,I have shown that it is not true for normed spaces because we can take the sup norm $||.||$ on $C[0,1]$ and we can take the function $f(x)=e^x$ and approximate it by a sequence of polynomials,but $e^x$ is not in the subspace spanned by those polynomials. But I could not find an easy example for infinite dimensional inner product space,can someone give an example of an infinite dimensional inner product space,where best approximation does not exist?
Does the following example work $V$ be the vector space of all eventually zero sequences in $\mathbb R$ and define the inner product $<\sigma,\mu>=\sum \sigma(n)\mu(n)$ and take the subspace $W={\sigma_n(k):n>1}$ where $\sigma_n=e_1+e_n$ where $e_n(k)=\delta_{n,k}$,consider $e_1$,does it have a best approximation in $W$?