Re: When the Gilbert space with measure is separable

On 29 Mar 2007 15:04:37 -0700, "Kostyantyn Yusenko"
<kay.math@xxxxxxxxx> wrote:


Robert Israel íàïèñàâ:


I'm not sure what you mean by L_2(H,m). I can think of two plausible
1) m is a measure on H, and you're dealing with scalar-valued functions
f on H where ||f||^2 = int_H |f(x)|^2 dm(x) < infty
2) m is a measure on some measurable space X, and you're dealing with
H-valued functions f on X where ||f||^2 = int_X ||f(x)||^2 dm(x) < infty.

In either case, you certainly want H to be separable. Whether m is
a probability measure is not really relevant, but you do want it to be
sigma-finite.
--


Well. In fact I meant the case 1). I will try to state the problem
more sharp.
If we are given measurable space X (not neccesarily Hilbert space) and
separable measure m,

What is a "separable" measure?

than the space L_2(X,m) (i.e. the space of scalar-valued functions f
on X that are summable with square) is separable.

And the question is if X is Hilbert space, are there some weaker
conditions on measure m for the space L_2(H,m) to be separable.


************************

David C. Ullrich
.

Relevant Pages