I am self studying Kreyszig book of functional analysis and couldn't solve following problem which is 8 th of section 3.4
Problem is
Show that an element $x$ of an inner product space $X$ cannot have too many Fourier coefficients $\langle x, e_k\rangle$ which are big, $\langle e_k\rangle$ is orthonormal sequence. Precisely, show that number $ n_m $ of $\langle x, e_k\rangle$ such that $| \langle x, e_k\rangle | > 1/m$ must satisfy $ n_m < m^2 ||x||^2.$
I tried using Bessel's inequality but I don't know how to exactly get $ n_m < m^2 ||x||^2.$
Best Answer
Let $S_m=\{e_k\mid\frac1m\lt |\langle x,e_k\rangle |\} $. Assume we have $n_m$ distinct elements in $S_m $, say $e_{l_1},\ldots,e_{l_{n_m}} $. Then $$\frac {n_m}{m^2}\lt \sum_{k=1}^{n_m} |\langle x,e_{l_k}\rangle |^2\leq \|x\|^2.$$