# Showing that an orthonormal set becomes a basis for the Hilbert space

This is an exercise from Folland Real Analysis Chapter 8 that I am stuck at. I am actually stuck at (b). I succeeded in showing that $$H_a$$ is a Hilbert space and the given set is an orthonormal set of $$H_a$$. However, I cannot show that the given set becomes a basis. I tried to apply the Stone Weierstrass Theorem; but the collection of finite linear combinations of the elements of the given set does not seem to form an algebra. The multiplication of $$\sqrt(2a)sinc(2ax-k)$$ and $$\sqrt(2a)sinc(2ax-k')$$ for arbitrary integers $$k$$ and $$k'$$ does not seem to be expressible as a finite linear combination of the given set.... I cannot find a way through. Could anyone please help me?

Based on part (a), I think they want you to use the fact that $$\hat{f} = \chi_{[-a,a]} \hat{f}$$ and Plancherel's theorem (i.e. the density of Fourier coefficients).