Hilbert space

Orthonormal dense basis

Let be a Hilbert space. An orthonormal dense basis¹ is an Orthonormal set and dense basis of . fun

Main theorem

If is a countable Orthonormal set, the following are equivalent 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

• Parseval’s relation allows the expansion of arbitrary inner products.

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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. ↩