Hilbert space and infinite norm vectors

Click For Summary

Discussion Overview

The discussion revolves around the definition and properties of Hilbert spaces, specifically whether a Hilbert space can be defined where vectors have infinite norms. The conversation also explores the concept of a vector space with an extended norm and its completeness regarding Cauchy sequences.

Discussion Character

  • Debate/contested
  • Conceptual clarification

Main Points Raised

  • Some participants assert that a Hilbert space cannot include vectors with infinite norms, citing the definition of Hilbert spaces.
  • One participant reformulates the question to consider a vector space with an extended norm and inquires about the completeness of this space regarding Cauchy sequences.
  • Another participant suggests looking into the extended complex plane and the Stone-Chech compactification of a Banach space as relevant concepts.

Areas of Agreement / Disagreement

Participants generally agree that a Hilbert space cannot be defined with infinite norm vectors, but there is ongoing debate regarding the properties of a vector space with an extended norm and its completeness.

Contextual Notes

The discussion does not resolve the implications of defining an extended norm or the completeness of the proposed vector space, leaving these aspects open for further exploration.

seratend
Messages
318
Reaction score
1
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?

Thanks in advance,

Seratend.
 
Physics news on Phys.org
It won't be a hilbert space, by definition.
 
matt grime said:
It won't be a hilbert space, by definition.

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?

Thanks in advance
Seratend.
 
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:
matt grime said:
Yes, look up the extended complex plane. It is even compact. See also the Stone-Chech compactification of a Banach Space, for instance.

Thanks a lot.

Seratend.
 

Similar threads

Undergrad Complete sets and complete spaces
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Understanding Hilbert Vector Spaces
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad About the definition of vector space of infinite dimension
  • · Replies 18 ·
Replies
18
Views
4K
Graduate Shauder basis for Hilbert spaces
  • · Replies 0 ·
Replies
0
Views
1K
Undergrad ##L^2## square integrable function Hilbert space
  • · Replies 11 ·
Replies
11
Views
4K
Undergrad Do Two Eigenvectors Form a Hilbert Space with Their Inner Product?
  • · Replies 2 ·
Replies
2
Views
3K
MHB What Makes Hilbert Space So Confusing?
  • · Replies 4 ·
Replies
4
Views
3K
Undergrad Shouldn't this definition of a metric include a square root?
  • · Replies 8 ·
Replies
8
Views
3K
Graduate Eigenvectors and matrix inner product
  • · Replies 1 ·
Replies
1
Views
3K
Insights Why Vector Spaces Explain The World: A Historical Perspective
  • · Replies 0 ·
Replies
0
Views
9K