
Fourier Coefficients
Dear Colleagues,
Could you please help me in solving the following problem:
Show that an element $\displaystyle x$ of an inner product space $\displaystyle X$ cannot have "too many" Fourier coefficients $\displaystyle <x, e_{k}>$ which are "big"; here, $\displaystyle (e_{k})$ is a given orthonormal sequence; more precisely, show that the number $\displaystyle n_{m}$ of $\displaystyle <x, e_{k}>$ such that $\displaystyle <x, e_{k}>>1/m$ must satisfy $\displaystyle n_{m}<m^{2}x^{2}$.
In fact the most important part in my opinion is how we can show that $\displaystyle n_{m}$ is finite?
Best Regards.

Use Parsevals' Identity.
Parseval's identity  Wikipedia, the free encyclopedia
we know that the sum
$\displaystyle \sum_{n=1}^{\infty}<x,e_n>$
Is finite.
Since the series converges we know that the limit
$\displaystyle \lim_{n \to \infty}<x,e_n>=0$

Thank you for your reply. In fact, Parseval's identity does not hold here since $\displaystyle (e_{k})$ is an orthonormal sequence not basis in the inner product space $\displaystyle X$.
Regards.
