Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

What do you call a maximal orthonormal set?

  1. Jan 19, 2008 #1


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    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: Jan 20, 2008
  2. jcsd
  3. Jan 19, 2008 #2
    It's called a Hilbert subset. All infinite-dimensional inner product spaces possess 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: Jan 20, 2008
  4. Jan 19, 2008 #3
    Or maybe try here...

    http://mathworld.wolfram.com/HilbertBasis.html" [Broken]

    and here...

    http://mathworld.wolfram.com/OrthonormalBasis.html" [Broken]
    Last edited by a moderator: May 3, 2017
  5. Jan 20, 2008 #4


    User Avatar
    Science Advisor
    Homework Helper

    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.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook