Special Properties of Hilbert Spaces?

Click For Summary
SUMMARY

The discussion centers on the unique properties of Hilbert Spaces, particularly their completeness and the relationship between inner products and norms. A Hilbert Space, denoted as H, is defined such that the norm is generated by an inner product, satisfying the Polarization Identity and the Parallelogram Law. The key distinction is that Hilbert Spaces are complete with respect to the norm induced by the inner product, meaning every Cauchy sequence converges within the space. This completeness is crucial for various mathematical applications, including the Riesz-Fischer representation theorem, which relies on the existence of unique projections.

PREREQUISITES
  • Understanding of inner product spaces and their properties
  • Familiarity with Cauchy sequences and convergence
  • Knowledge of the Riesz-Fischer representation theorem
  • Basic concepts of metric spaces and topology
NEXT STEPS
  • Study the properties of L^p spaces, particularly for p ≠ 2
  • Explore the implications of the Polarization Identity in functional analysis
  • Investigate the completeness theorem and its applications in various mathematical contexts
  • Learn about projection operators and their significance in Hilbert Spaces
USEFUL FOR

Mathematicians, physicists, and students in advanced mathematics who are studying functional analysis, particularly those interested in the applications of Hilbert Spaces in various fields such as probability and quantum mechanics.

  • #31
Oh, I see what you were getting at.

To get a counter-example, multiply the inner product norm by a small constant. It's still a norm. And note that if you inner product something with itself, you have equality in the Cauchy-Schwartz inequality. The shrunken square norm will be less than the regular square norm.
 
Physics news on Phys.org
  • #32
And the counter-example is a norm that actually does come from some inner product. Just rescale the inner product.
 
  • #33
I should add that the same trick works for a norm that doesn't come from an inner product, if you want it to work for one of those. If it did obey Cauchy-Schwarz, you could re-scale it so that it didn't for any given inner product.
 
  • #34
O.K, good point.
 

Similar threads

Graduate Reproducing Kernel Hilbert Spaces
  • · Replies 3 ·
Replies
3
Views
2K
Insights Hilbert Spaces And Their Relatives - Operators
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad ##L^2## square integrable function Hilbert space
  • · Replies 11 ·
Replies
11
Views
3K
Bounded operators on Hilbert spaces
  • · Replies 43 ·
2
Replies
43
Views
5K
Graduate Hilbert spaces and kets "over" manifolds
  • · Replies 10 ·
Replies
10
Views
3K
Undergrad Understanding Hilbert Vector Spaces
  • · Replies 3 ·
Replies
3
Views
2K
Graduate The polarization identity in Hilbert space
  • · Replies 2 ·
Replies
2
Views
7K
Graduate How to derive the quantum detailed balance condition?
  • · Replies 0 ·
Replies
0
Views
1K
Undergrad Complete sets and complete spaces
  • · Replies 2 ·
Replies
2
Views
2K
Uncovering the Relationship Between Norms and Hilbert Spaces
  • · Replies 1 ·
Replies
1
Views
1K