A compact operator is a type of linear operator that maps elements from one Banach space to another, with the special property that 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 Hilbert spaces and are important in various applications, including quantum mechanics and partial differential equations. Their spectral properties are well-studied, leading to significant results in mathematical analysis.