Compact operators and approximation properties

For Banach spaces X and Y, a bounded linear operator T is compact when the closure of T(B_X) is norm compact in Y. Equivalently, images of every bounded sequence have a norm-convergent subsequence. Bounded finite-dimensional images have compact closure, so bounded finite-rank operators are compact. The identity on an infinite-dimensional Banach space is not compact, since Riesz’s lemma constructs separated unit vectors. Hence boundedness of an operator, or boundedness of its image, is much weaker than compactness.

The operator-norm limit of compact maps into Banach Y is compact. A finite net for one approximating image, together with a uniform norm-error estimate on the unit ball, gives a finite net for the limiting image; completeness turns total boundedness into compact closure. A compact map also sends a weakly convergent sequence to a norm-convergent sequence with the corresponding image limit. Otherwise a subsequence staying away from that limit would have a compact image subsequence with another norm limit, contradicting weak continuity of a bounded linear map.

A Banach space Y has the approximation property when for every norm-compact subset K and every positive error, some bounded finite-rank map R:Y→Y approximates the identity uniformly on K. This definition does not require one bound on all the chosen operator norms. Requiring such a common bound gives the stronger bounded approximation property. The approximation is on compact sets, not uniformly on the entire unit ball; in infinite dimension the latter would incorrectly make the identity compact.

If the target Y has this property and T:X→Y is compact, apply it to the compact closure of T(B_X). Then RT has finite rank and approximates T in operator norm. This proves the stated sufficient condition with the property placed on the target, without an unjustified claim about an arbitrary domain. Hilbert spaces have the property through finite-dimensional orthogonal projections, and spaces with a Schauder basis have the bounded approximation property. Enflo’s counterexample shows that not every Banach space has even the unbounded approximation property, so the general converse to finite-rank compactness fails.

Coordinate projections on ℓ² converge strongly to the identity: every fixed square-summable vector has a small tail. Their operator-norm error from the identity is nevertheless one, because a unit vector beyond the cutoff remains untouched by the projection. Adding diagonal weights that tend to zero changes the situation. The largest omitted weight then controls the truncation error uniformly over the entire input unit ball, giving a compact diagonal operator as an operator-norm limit of finite-rank maps.

Schauder’s theorem states that a bounded operator between Banach spaces is compact exactly when its adjoint is compact; this theorem does not require an approximation property. It should not be proved by assuming finite-rank approximation in arbitrary spaces, since that would use a false general premise. Such distinctions separate three ideas often conflated in applications: compact image geometry, strong convergence of a chosen approximation, and operator-norm convergence. An explicit tail bound can certify the last property in a concrete model even when the abstract compactness argument does not specify an approximation.