Orthonormal dense basis

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

Main theorem

If $E = {| 𝑒 𝑛 ⟩} \infty 𝑛 = 1$ is a countable Orthonormal set, the following are equivalent #m/thm/anal/fun

$E$ is an orthonormal dense basis of $𝑋$
$E ⟂ = {0}$
$| 𝑥 ⟩ = \sum \infty 𝑛 = 1 | 𝑒 𝑛 ⟩ ⟨ 𝑒 𝑛 | 𝑥 ⟩$ for all $𝑥 \in 𝑋$

Proof

Assume $E$ is an orthonormal dense basis of $𝑋$ and let $𝑌 = s p a n E$ . Note that $E ⟂ = 𝑌 ⟂$ by ^S6. Let $𝑥 \in 𝑌 ⟂$ . Now by the density of $𝑌$ there exists a sequence $(| 𝑦 𝑖 ⟩) \infty 𝑖 = 1$ in $𝑌$ such that $l i m 𝑛 \to \infty | 𝑦 𝑛 ⟩ = 𝑥$ . But since the inner product is continuous

⟨ 𝑥 | 𝑥 ⟩ = l i m 𝑛 \to \infty ⟨ 𝑥 | 𝑦 𝑘 ⟩ = 0

whence $\Span so ^O1 implies ^O2.

Now assume $E ⟂ = {0}$ . Let

| 𝑥 𝑘 ⟩ = 𝑘 \sum 𝑗 = 1 | 𝑒 𝑗 ⟩ ⟨ 𝑒 𝑗 | 𝑥 ⟩

We will show that $l i m 𝑘 \to \infty | 𝑥 𝑘 ⟩ = | 𝑥 ⟩$ .

Properties

Parseval's relation allows the expansion of arbitrary inner products.

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

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

Orthonormal dense basis

Main theorem

Properties

Footnotes