# Sequence in Hilbert space,example

1. Jul 28, 2012

### shotgun07

Hi. Im working on some exercises but I couldt find any clue for this one:

Find a bounded sequence (as like the norm) in l^2 Hilbert space,that weakly converges to 0 (as like the weak topology)
but doesn`t have any convergent subsequences (as in strong topology).
Could someone help me?

2. Aug 7, 2012

### Bacle2

I think the unit cube in Hilbert space l^2:{ei:=δij}, i.e., ei

has a 1 in the i-th position and is 0 otherwise, is an example: for one thing, it does

not have a finite ε-net for ε <1/2, say, since the l2-distance is √2, so that it

cannot have convergent subsequences -- the √2 distance is a barrier to being Cauchy.

3. Aug 8, 2012

### Bacle2

I forgot the second part, about the sequence converging weakly:

first , in l2:

We need to show:

Limn→∞ <en, an>=<0,an>=0

But notice that the product on the left equals the n-th term of a sequence an

in l2. Since an is square-summable, it is Cauchy, so that

its n-th term goes to zero as n→ ∞.

But this sequence {en}=(δij)j=1,...,∞ converges weakly

to zero in _any_ Hilbert space:

Let h be an element in any Hilbert space H. Then, since {en} is a maximal

orthonormal set, it is a Hamel basis for H . Then h is a finite linear combination of the

basis elements in {en}. Let ek be the largest index in the

linear combination. Then, when n>k , < en,h>=0.

h=Ʃ