# Convergence in Hilbert space question

1. Feb 1, 2008

### quasar987

1. The problem statement, all variables and given/known data
Is it true/possible to show that in a Hilbert space, if z_n is a sequence (not known to converge a priori) such that (z_n,y)-->0 for all y, then z_n-->0 ?

3. The attempt at a solution

I've shown that if z_n converges, then it must be to 0. But does it converge?

2. Feb 1, 2008

### Dick

Consider an orthonormal basis of the Hilbert space, {e_n} and let the sequence be the basis vectors. Now (e_n,y) is the nth component of y, which must tend to zero as n goes to infinity for any y. Yet e_n clearly does not converge. So no, you can't show that.

3. Feb 2, 2008

### quasar987

Good example!