Let $\{x_k\}$ be a collection of vectors in a Hilbert space. We take any $x\in H$. The symbol $\langle .,.\rangle$ denote the inner product. The question is as follows. I have to apply the Parseval's theorem to
$$\sum_{k=1}^\infty |\langle x,x_k\rangle|^2$$
to obtain a relationship with the Fourier transforms. How should I proceed? What is the result?
Thank you very much.
First we shall assume that the basis are orthonormal, i.e. $$\left\langle {{x_i},{x_j}} \right\rangle = \left\{ \begin{array}{l}1,i = j\\0,i \ne j\end{array} \right.$$ now, merely expanding the arbitrary vector in terms of these basis yields $$x = \sum\limits_{i = 1}^\infty {\left\langle {x,{x_i}} \right\rangle {x_i}}$$ Now, the "energy" of the vector (it's 2-norm) could be calculated as$$\left\langle {x,x} \right\rangle = \sum\limits_{i = 1}^\infty {\sum\limits_{j = 1}^\infty {\left\langle {x,{x_j}} \right\rangle \left\langle {x,{x_i}} \right\rangle \left\langle {{x_i},{x_j}} \right\rangle } } = \sum\limits_{i = 1}^\infty {{{\left| {\left\langle {x,{x_i}} \right\rangle } \right|}^2}}$$which is probably your desired result. In text, the result is that the 2-norm of the vector (it's length) is the sum of square of it's expansion coefficient (which is quite obvious in Euclidean geometry).