Sequence in Hilbert space,example

[shotgun07]

Hi. I`m working on some exercises but I could`t 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?

 

[Bacle2]

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

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.

 

[Bacle2]

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

first , in l²:

We need to show:

Lim_n→∞ <e_n, a_n>=<0,a_n>=0

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

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

its n-th term goes to zero as n→ ∞.

But this sequence {e_n}=(δ_ij)j=1,...,∞ converges weakly

to zero in _any_ Hilbert space:

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

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

basis elements in {e_n}. Let e_k be the largest index in the

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

h=Ʃ