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.