Hilbert space

Orthonormal dense basis

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

Main theorem

If E ={|𝑒𝑛⟩}∞𝑛=1 is a countable Orthonormal set, the following are equivalent fun

1. E is an orthonormal dense basis of 𝑋
2. E⟂ ={0}
3. |𝑥⟩ =∑∞𝑛=1|𝑒𝑛⟩⟨𝑒𝑛|𝑥⟩ for all 𝑥 ∈𝑋

Proof

Assume E is an orthonormal dense basis of 𝑋 and let 𝑌 =span⁡E. Note that E⟂ =𝑌⟂ by ^S6. Let 𝑥 ∈𝑌⟂. Now by the density of 𝑌 there exists a sequence (|𝑦𝑖⟩)∞𝑖=1 in 𝑌 such that lim𝑛→∞|𝑦𝑛⟩ =𝑥. But since the inner product is continuous

⟨𝑥|𝑥⟩=lim𝑛→∞⟨𝑥|𝑦𝑘⟩=0

whence $\Span so ^O1 implies ^O2.

Now assume E⟂ ={0}. Let

|𝑥𝑘⟩=𝑘∑𝑗=1|𝑒𝑗⟩⟨𝑒𝑗|𝑥⟩

We will show that lim𝑘→∞|𝑥𝑘⟩ =|𝑥⟩.

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. ↩