2
$\begingroup$

Say we have an orthonormal basis $\{e_n\}$ for a infinite Hilbert Space $H$. I want to prove that any vector $x=\sum_{n=1}^\infty\langle x, e_n\rangle e_n$. I don't know where to start. Could I have any help?

  • 0
    Thank you. My mistake. I didn't read over this after I posted.2012-09-01

2 Answers 2

1

What is the meaning of $ x=\sum_{n=1}^\infty \langle x,e_n\rangle e_n? $ This is $ x=\lim_{m\to\infty}\sum_{n=1}^m \langle x,e_n\rangle e_n, $ or in other words $ \left\|x-\sum_{n=1}^m \langle x,e_n\rangle e_n\right\|\to 0, $ as $m\to\infty$. Now recall $\|y\|^2=\langle y,y\rangle$, and try to show that the preceding limit goes to $0$.

1

Consider the set of all vectors of that form. Prove that it is a (closed) subspace, and that its orthogonal is zero. Then it has to be all of $H$.