1. ## Fourier Coefficient

Please can you guys help me to solve the following question. I couldnt even get the question.
Q. Show that an element x of an inner product space X cannot have "too many" Fourier coefficeints <x,e_k> which are "big"; here, (e_k) is given orthonormal sequence; more precisely, show that number n_m of <x,e_k> such that |<x,e_k>| > 1/m must satisfy n_m < m^2 ||x||^2.

2. Originally Posted by kinkong
Please can you guys help me to solve the following question. I couldnt even get the question.
Q. Show that an element x of an inner product space X cannot have "too many" Fourier coefficeints <x,e_k> which are "big"; here, (e_k) is given orthonormal sequence; more precisely, show that number n_m of <x,e_k> such that |<x,e_k>| > 1/m must satisfy n_m < m^2 ||x||^2.
Is this a Hilbert space? Is that orthonormal basis a Hamel or Schauder basis?

3. yes it is a Hilbert space...i think it is Hilbert basis too...i couldnt even understand the question properly

4. Originally Posted by kinkong
Please can you guys help me to solve the following question. I couldnt even get the question.
Q. Show that an element x of an inner product space X cannot have "too many" Fourier coefficeints <x,e_k> which are "big"; here, (e_k) is given orthonormal sequence; more precisely, show that number n_m of <x,e_k> such that |<x,e_k>| > 1/m must satisfy n_m < m^2 ||x||^2.
Hint: Bessel's inequality.

5. Please can you help me to solve it. i coult get it from your hint. Thanks

6. Originally Posted by kinkong
Please can you help me to solve it.
Bessel's inequality says that $\sum|\langle x,e_k\rangle|^2 \leqslant \|x\|^2$. The aim now is to prove by contradiction that there are fewer than $m^2\|x\|^2$ values of k for which $|\langle x,e_k\rangle|>1/m$.

So suppose that there are at least $m^2\|x\|^2$ values of k for which $|\langle x,e_k\rangle|>1/m$. The the sum $\sum|\langle x,e_k\rangle|^2$ will be greater than _____ ?

7. Originally Posted by Opalg
Bessel's inequality says that $\sum|\langle x,e_k\rangle|^2 \leqslant \|x\|^2$. The aim now is to prove by contradiction that there are fewer than $m^2\|x\|^2$ values of k for which $|\langle x,e_k\rangle|>1/m$.

So suppose that there are at least $m^2\|x\|^2$ values of k for which $|\langle x,e_k\rangle|>1/m$. The the sum $\sum|\langle x,e_k\rangle|^2$ will be greater than _____ ?
Will it be greater than ||x||^2??