Re-reading your question, I think that I see what you are asking.
Per @Andrea Ferretti's comments, you have to be careful to distinguish between einx and spaneinx. You certainly are interested the latter. Sorry if my comments were sloppy and confusing above.
So, I think that the it goes like this:
From some corollary of Stone-Weirstrauss you can show that spaneinx is dense in C(mathbbS1) with the supremum norm. Because we know that C(mathbbS1)hookrightarrowL2([0,1]) has its image a dense subset of L2([0,1]) and we know that if fntof in the supremum topology on C(mathbbS1), then the images also converge in L2([0,1]).
Thus, by this reasoning, for finL2([0,1]) we can find fninspaneinx such that fn=L2([0,1]). Lets write
fn=sumkinmathbbZc(n)keikx
where all but finitely many of the c(n)k are zero (this is because in the span of infinitely many objects we only take a finite number of them to add together)
Now, what I think you are asking is: what can we say about the coefficients c(n)k? The answer is that they converge to the k-th Fourier coefficient of f as ntoinfty because
hatf(k)=langlef,eikxrangle=limntoinftylanglefn,eikxrangle=limntoinftyc(n)k
In fact if c(n)k are arbitrary complex numbers, defining fn as above, we see that
Vertf−fnVertL2=sumkinmathbbZ|hatf(k)−c(n)k|2
assuming convergence. Thus, if (c(n)k)kto(hatf(k))k as ntoinfty in ell2(mathbbZ) then fntof in L2, which is a pretty weak condition.
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