6.3 Compact \(\Leftrightarrow \) approximable on Hilbert spaces

Theorem 6.9

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Every compact \(S \in \mathcal{L}(H_1, H_2)\) between Hilbert spaces is approximable.

Proof ▼

The Schmidt representation (Theorem 6.5) gives singular values \(\sigma _n \to 0\) which, by Eckart–Young (Theorem 6.6), equal the approximation numbers \(a_n(S)\); hence \(a_n(S) \to 0\). Equivalently, the SVD truncations are finite-rank operators converging to \(S\) in operator norm.

Theorem 6.10

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On Hilbert spaces, \(S\) is approximable iff \(S\) is compact.

Proof ▼

Combine the two implications: approximable \(\Rightarrow \) compact is Theorem 6.3, and compact \(\Rightarrow \) approximable is Theorem 6.9.