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Confused with Complex Vector Space notation

I've attached a picture with the problem statement. I've done some least squares problems with matrices, but I don't think that will work here.

I thought I'd just find the orthogonal projection of f(x) onto W directly, but in order to do that I need basis vectors for W.

I don't really understand what the vector space W is here. Is it a subspace of V, where V is all continuous functions with a function value between -pi and pi? Or is it the input in e^ix which is between -pi and pi?

If I could get some help clarifying that vector space notation, I believe I will be able to solve the problem.

Thanks!

Re: Confused with Complex Vector Space notation

W is a vector space of functions, the smallest containing the functions x->e^ix and x->e^-ix (all functions that can be written ae^ix+be^-ix), so it's a subspace of V.

Re: Confused with Complex Vector Space notation

All right, thank you for replying! But is V the vector space of all complex functions where the input is between -pi and pi?

Re: Confused with Complex Vector Space notation

Yes, V the vector space of all continuous complex functions where the input is between -pi and pi