# What do you call a maximal orthonormal set?

1. Jan 19, 2008

### quasar987

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. Jan 19, 2008

### mathboy

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
3. Jan 19, 2008

### Ynaught?

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
4. Jan 20, 2008

### morphism

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.