# Hilbert space

1. Dec 8, 2008

### dirk_mec1

1. The problem statement, all variables and given/known data
http://img168.imageshack.us/img168/5042/48390466ny3.png [Broken]

2. Relevant equations

A orthonormal system is if $$f_i \cdot f_j = 0$$ for all $$i \neq j$$ and if $$||f_i||=1$$

A sequence contained in $$l^{\infty}$$ is a bounded sequence.

http://img99.imageshack.us/img99/1840/67874379ps9.png [Broken]

3. The attempt at a solution

My guess is that I have to use theorem 9.3 but I don't understand the notation. <x,e_n> is x just a number?

Last edited by a moderator: May 3, 2017
2. Dec 8, 2008

### HallsofIvy

Yes, $<x, e_n>$ is just a number and so $<x, e_n>\lambda_n$, for each n, is just a number. Apply your theorem 9.3 with the $\lambda_n$ in that theorem equal to the $<x, e_n>\lambda_n$ here.

3. Dec 8, 2008

### tiny-tim

Hi dirk_mec1!

I haven't read the whole problem,

but just answering the last sentence:

x is a vector, just like e_n, and the inner product, <x,e_n> , is a number.