6.1 Approximable operators
\(S \in \mathcal{L}(X, Y)\) is approximable if \(a_n(S) \xrightarrow {n \to \infty } 0\), equivalently if \(S\) is the operator-norm limit of a sequence of finite-rank operators.
The approximable operators form a closed two-sided operator ideal: a subspace of \(\mathcal{L}(X, Y)\), closed under pre/post-composition with bounded operators, closed in operator-norm topology, and containing all finite-rank operators.
The equivalence with being a norm-limit of finite-rank operators follows by choosing near-optimal rank-\(\le n\) approximants \(L_n\) with \(\| S - L_n\| {\lt} a_n(S) + \tfrac {1}{n+1}\), and conversely by squeezing \(a_n(S) \le \| S - L_n\| \). Stability under addition combines rank-\(\le \lfloor n/2 \rfloor \) and rank-\(\le \lceil n/2 \rceil \) approximants of the two summands, using that ranks are subadditive; stability under composition and closedness in operator norm are the axioms (S3) and (S2) of \(a_n\) passed through the limit.
Every approximable operator \(S : X \to Y\) from a normed space \(X\) to a Banach space \(Y\) over a locally compact field \(\mathbb {K}\) is compact. In particular every operator of finite rank is compact, since finite-rank operators are approximable (Proposition 6.2).
Each finite-rank approximant \(L_n\) corestricts to its range, a finite-dimensional subspace of \(Y\), which is locally compact because \(\mathbb {K}\) is; an operator into a locally compact space is compact, and post-composition with the inclusion preserves compactness. Since compactness passes to operator-norm limits, \(S = \lim _n L_n\) is compact.