What do you call a maximal orthonormal set?

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Discussion Overview

The discussion revolves around the terminology used for a maximal orthonormal set in inner product spaces, particularly in the context of Hilbert spaces. Participants explore different names and definitions, as well as the implications of these terms in both finite and infinite-dimensional spaces.

Discussion Character

  • Debate/contested
  • Conceptual clarification

Main Points Raised

  • One participant inquires about the specific name for a maximal orthonormal set, mentioning that their professor referred to it as a "Hilbert basis."
  • Another participant suggests the term "Hilbert subset," asserting that all infinite-dimensional inner product spaces possess such a set, which can be proven using Zorn's Lemma.
  • A third participant provides links to MathWorld for further reading on "Hilbert Basis" and "Orthonormal Basis."
  • A later reply agrees that a maximal orthonormal set need not be a basis in arbitrary inner product spaces but states that in Hilbert spaces, it is typically referred to as a Hilbert space basis or orthonormal basis.

Areas of Agreement / Disagreement

Participants express differing views on the appropriate terminology, with no consensus reached on a single term for maximal orthonormal sets. Some argue for "Hilbert basis," while others contest this by suggesting "Hilbert subset" or "orthonormal basis" in specific contexts.

Contextual Notes

There is ambiguity regarding the definitions and implications of the terms used, particularly in distinguishing between finite and infinite-dimensional spaces. The discussion highlights the need for clarity in terminology within the context of inner product spaces.

quasar987
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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.
 
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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.
 
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Or maybe try here...

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

and here...

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

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