Proving Orthonormality & Boundedness of Vector Sequence

[dirk_mec1]

Homework Statement

http://img168.imageshack.us/img168/5042/48390466ny3.png

Homework 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

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?

 

[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.

 

[tiny-tim]

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?

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.