Approximation property

In mathematics, specifically functional analysis, a Banach space is said to have the approximation property (AP), if every compact operator is a limit of finite-rank operators. The converse is always true.

Every Hilbert space has this property. There are, however, Banach spaces which do not; Per Enflo published the first counterexample in a 1973 article. However, much work in this area was done by Grothendieck (1955).

Later many other counterexamples were found. The space ${\displaystyle {\mathcal {L}}(H)}$ of bounded operators on an infinite-dimensional Hilbert space ${\displaystyle H}$ does not have the approximation property. The spaces ${\displaystyle \ell ^{p}}$ for ${\displaystyle p\neq 2}$ and ${\displaystyle c_{0}}$ (see Sequence space) have closed subspaces that do not have the approximation property.

A locally convex topological vector space X is said to have the approximation property, if the identity map can be approximated, uniformly on precompact sets, by continuous linear maps of finite rank.

For a locally convex space X, the following are equivalent:

where ${\displaystyle \operatorname {L} _{p}(X,Y)}$ denotes the space of continuous linear operators from X to Y endowed with the topology of uniform convergence on pre-compact subsets of X.

If X is a Banach space this requirement becomes that for every compact set ${\displaystyle K\subset X}$ and every ${\displaystyle \varepsilon >0}$, there is an operator ${\displaystyle T\colon X\to X}$ of finite rank so that ${\displaystyle \|Tx-x\|\leq \varepsilon }$, for every ${\displaystyle x\in K}$.

Some other flavours of the AP are studied: