A compact operator is a type of linear operator that maps elements from one Banach space to another, while having a special property: it takes bounded sets to relatively compact sets. This means that the image of any bounded set under a compact operator has compact closure, which is crucial in functional analysis. Compact operators can be thought of as generalizations of matrices in infinite-dimensional spaces. They often arise in the study of differential equations and integral equations, where they help in understanding the behavior of solutions. Examples include the Hilbert-Schmidt operators and Fredholm operators, which are important in various mathematical applications.