Compact operator

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

has compact closure in $Y$.

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