1. Fourier Coefficients

Dear Colleagues,

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.

2. 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

$\displaystyle \lim_{n \to \infty}<x,e_n>=0$

3. 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.

4. Sorry You need Bessel's inequality then

Bessel's inequality - Wikipedia, the free encyclopedia