As we know that

If $H$ is a Hilbe$\{e_n\}$ be an orthonormal sequence then for all $x\in H$ $$\|x\|^2\ge \sum_{k=1}^\infty|\langle x,e_k\rangle|^2$$ and the infinite sum converges to $$\sum_{k=1}^\infty\langle x,e_k \rangle e_k. $$

I was thinking for that if we don't have an orthonormal basis then can we give an example of such sequence which satisfies $\|x\|^2=\sum_{k=1}^\infty| \langle x,e_k\rangle|^2\ \ $ for all $x\in H$.

  • 1
    $\begingroup$ What do you mean by 'don't have an orthonormal basis'? Do you mean any sequence or a Schauder basis that is not orthornomal? $\endgroup$ – copper.hat Dec 21 '16 at 19:43
  • $\begingroup$ Any sequence or a Schauder basis anything. $\endgroup$ – Sachchidanand Prasad Dec 21 '16 at 19:47

If you have the Parseval equality, then the $e_n$ are orthogonal because $$ \|e_n\|^2 = \sum_{k=1}^{\infty}|\langle e_n,e_k\rangle|^2=\|e_n\|^2+\sum_{k\ne n}|\langle e_n,e_k\rangle|^2. $$


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.