Re: Hilbert Scmidt norm need help with proof

In article <1257e2f8.0504280819.52d6902d@xxxxxxxxxxxxxxxxxx>,
egilbae@xxxxxxxxxxx wrote:

> If {en} and {fn} are complete orthonormal sequences in Hilbertspaces
> H,K respectivly, A:H->K linear and continuous then
> sum[1->oo](||Aen||^2) = sum[1->oo](||A*fn||^2) , A* = adjoint of A.
> Why is this true?
>
> I have been stuck with this for two hours, but there might be some
> very easy way to prove it, hope someone can help me.

sum_n ||Aen||^2 = sum_n sum_m |<Aen,fm>|^2

= sum_n sum_m |<en,A*fm>|^2 = sum_m sum_n |<en,A*fm>|^2

= sum_m ||A*fm||^2
.