Re: separable Hilbert space

Edward Green wrote:

> michaeld wrote:
> > David Macmanus wrote:
> > > Can anyone give an example of a Hilbert space that isn't separable?
> > > Separable Hilbert spaces have orthonormal bases which are either finite
> > > or infinite, but countable. So I guess I'm wondering what's a space that
> > > has an uncountable basis set of vectors.
> > > Thanks,
> > > David.
> >
> > Let I be some big index set. Let H be the set of all complex valued
> > sequences v indexed by I, i.e. v: I->C, such that:
> >
> > sum |v(i)|^2
> > I
> >
> > is finite. Note that if the sequence v is in H then this condition
> > implies that all but countably many of the v(i) must equal zero.
>
> Why? Did you mean to say that v(i) is limited in value to 0,1?

No. v(i) is a general complex number.

Note that for any e > 0 then the set S_e = {i in I | |v(i)| > e} is
finite - because each element of this set contributes at least e^2 to
the sum.

Now {i in I | |v(i)| > 0} is the union of S_e over e = 1,1/2,1/3,...
i.e. it is a countable union of countable sets, so is countable. So the
set of i such that v(i) != 0 is countable, if that sum is to have any
chance of being finite.

.

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