# Hilbert space and infinite norm vectors

1. Dec 17, 2004

### seratend

Quickly can we define a hilbert space (H, <,>) where the vectors of this space have infinite norm? (i.e. the union of finite + infinite norm vectors form a complete space).
If yes, can you give a link to a paper available on the web? If no, can you briefly describe why?

Seratend.

2. Dec 17, 2004

### matt grime

It won't be a hilbert space, by definition.

3. Dec 17, 2004

### seratend

Thanks, for the precision.

OK, so I try to reformulate my question,
Just take a vector space E. Where we define the extended norm as an application between the vectors of E and |R+U{+oO} plus the usual norm properties.
can we have the complete property (i.e. convergence of any cauchy sequence in this space) in this vector space?

Seratend.

4. Dec 17, 2004

### matt grime

Yes, look up the extended complex plane. It is even compact. See also the Stone-Chech compactification of a Banach Space, for instance.

Last edited: Dec 17, 2004
5. Dec 17, 2004

### seratend

Thanks a lot.

Seratend.