# Compact operators

## Definition

For a linear operator between Banach spaces, compactness is a natural strengthening of continuity (boundedness). A linear operator is compact if it sends the bounded subsets to relatively compact subsets.

## Properties

Since every relatively compact subspace (in a Banach space, or indeed in any metric space) is bounded, every compact operator is bounded. Instead of checking compactness on all bounded subsets it is sufficient to check it for a ball of one fixed radius: an operator is compact iff it sends the ball of unit radius to a relatively compact set.