What do you call a maximal orthonormal set?

  • Thread starter Thread starter quasar987
  • Start date Start date
  • Tags Tags
    Set
Click For Summary
SUMMARY

A maximal orthonormal set in an inner product space is commonly referred to as an "orthonormal basis" in the context of Hilbert spaces. While some sources may incorrectly label it as a "Hilbert basis," this term is not universally accepted, as a maximal orthonormal set does not always generate the vector space. The existence of a Hilbert set in infinite-dimensional inner product spaces can be established using Zorn's Lemma. For clarity, it is essential to differentiate between the terms "Hilbert basis" and "orthonormal basis" based on the context of the space being discussed.

PREREQUISITES
  • Understanding of inner product spaces
  • Familiarity with Hilbert spaces
  • Knowledge of Zorn's Lemma
  • Basic concepts of orthonormal sets
NEXT STEPS
  • Research the properties of Hilbert spaces and their bases
  • Study Zorn's Lemma and its applications in functional analysis
  • Explore the differences between orthonormal bases and maximal orthonormal sets
  • Investigate the implications of the Riesz Representation Theorem in Hilbert spaces
USEFUL FOR

Mathematicians, students of functional analysis, and anyone studying the properties of inner product spaces and Hilbert spaces will benefit from this discussion.

quasar987
Science Advisor
Homework Helper
Gold Member
Messages
4,796
Reaction score
32
Is there a specific name for a maximal orthonormal set in an inner product space? My professor called this a "Hilbert basis" (except the french translation of this). But wiki doesn't seem to know what a Hilbert basis is.
 
Last edited:
Mathematics news on Phys.org
It's called a Hilbert subset. All infinite-dimensional inner product spaces possesses a Hilbert set and this can be proven using Zorn's Lemma (a nice exercise).

Hilbert basis is definitely not the name, because a maximal orthonormal set of an infinite-dimensional inner product space does not necessarily generate the vector space.
 
Last edited:
Or maybe try here...

http://mathworld.wolfram.com/HilbertBasis.html"

and here...

http://mathworld.wolfram.com/OrthonormalBasis.html"
 
Last edited by a moderator:
mathboy is right in that a maximal orthonormal set of an arbitrary inner product space need not be a basis. However, in a Hilbert space (which, judging the flavor of your recent posts, is probably what you're working with) it is. In this case, these things are usually called Hilbert space bases or, more generally, orthonormal bases.
 

Similar threads

Undergrad Inner and Outer Product of the Wavefunctions
  • · Replies 8 ·
Replies
8
Views
4K
Undergrad QM Qubit state space representation by Projective Hilbert space
  • · Replies 18 ·
Replies
18
Views
1K
Undergrad Complete sets and complete spaces
  • · Replies 2 ·
Replies
2
Views
2K
High School My argument why Hilbert's Hotel is not a veridical Paradox
  • · Replies 8 ·
Replies
8
Views
3K
MHB Books to Learn Measure Theory Theory: Borel, Lebesgue, Cantor Set & More
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Proof that if T is Hermitian, eigenvectors form an orthonormal basis
  • · Replies 3 ·
Replies
3
Views
3K
Undergrad Orthonormal basis expression for ordinary contraction of a tensor
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Orthonormal Basis of Wavefunctions in Hilbert Space
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad How to calculate basis vectors from metric tensor (as a matrix)?
  • · Replies 6 ·
Replies
6
Views
865
Undergrad Orthogonal 3D Basis Functions in Spherical Coordinates
  • · Replies 3 ·
Replies
3
Views
4K