Convergence in Hilbert space question

[quasar987]

Homework Statement

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 ?

The Attempt at a Solution

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

 

[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.

 

[quasar987]

Good example!