 Compact operator

A bounded linear operator $T:X\to Y$ between normed spaces is a compact operator if the image of the unit ball has compact closure in $Y$. Equivalently, every bounded input sequence has a subsequence whose images converge in the target norm.

On ${\ell }^2$, $(Tx{)}_n=x_n/n$ is compact. Retaining its first $N$ coordinates gives a finite-rank operator, and the operator-norm error is $1/(N+1)$. By contrast, the identity on infinite-dimensional ${\ell }^2$ is not compact: it maps the standard unit vectors to themselves, and no subsequence is norm convergent. Compactness is therefore a stronger property than boundedness, reflecting a form of finite-dimensional approximability in this example.

In functional analysis, a compact operator is a specific type of linear operator that maps elements from one Banach space to another (or possibly to the same space) with properties similar to those of compact sets in finite-dimensional spaces. The concept of compact operators is crucial in the study of various problems in applied mathematics, quantum mechanics, and functional analysis. ### Definition Let \( X \) and \( Y \) be two Banach spaces.