Hilbert space

Orthonormal dense basis

Let be a Hilbert space. An orthonormal dense basis1 is an Orthonormal set and dense basis of . #m/def/anal/fun

Main theorem

If is a countable Orthonormal set, the following are equivalent #m/thm/anal/fun

  1. is an orthonormal dense basis of
  3. for all
Proof

Assume is an orthonormal dense basis of and let . Note that by ^S6. Let . Now by the density of there exists a sequence in such that . But since the inner product is continuous

whence $\Span so ^O1 implies ^O2.

Now assume . Let

We will show that .

Properties

#state/develop | #lang/en | #SemBr

Footnotes

  1. This is nonstandard terminology. Normally, this is just called an orthonormal basis, while the normal definition of a basis is relegated to Hammel basis. ↩