Compact operator - Imported Knowlpedia Content

A compact operator is a linear operator $T:X\to Y$ between normed vector spaces such that the image of the closed unit ball

B_X=\{x\in X:\|x\|\le 1\}

has compact closure in $Y$ .

Equivalent characterizations

Equivalently, for every bounded sequence $(x_n)$ in $X$ , the sequence $(Tx_n)$ has a subsequence that is a convergent sequence in $Y$ .

Remarks

In finite-dimensional normed spaces, every linear operator is compact; compactness becomes a restrictive and useful condition primarily in infinite-dimensional settings.

Examples

Any operator with finite-dimensional range (a “finite-rank” operator) is compact; for instance, a projection onto the span of finitely many vectors in a Hilbert space is compact.
Any linear map between finite-dimensional normed vector spaces is compact.