# Compact operator

An operator $A$ defined on a subset $M$ of a topological vector space $X$, with values in a topological vector space $Y$, such that every bounded subset of $M$ is mapped by it into a pre-compact subset of $Y$. If, in addition, the operator $A$ is continuous on $M$, then it is called completely continuous on this set. In the case when $X$ and $Y$ are Banach or, more generally, bornological spaces and the operator $A: X \to Y$ is linear, the concepts of a compact operator and of a completely-continuous operator are the same. If $A$ is a compact operator and $B$ is a continuous operator, then $A \circ B$ and $B \circ A$ are compact operators, so that the set of compact operators is a two-sided ideal in the ring of all continuous operators. In particular, a compact operator does not have a continuous inverse. The property of compactness plays an essential role in the theory of fixed points of an operator and in the study of its spectrum, which, in this case, has a number of ‘good’ properties.
Examples of compact operators are the Fredholm integral operator $$A(x) = \int_{a}^{b} K(t,s) ~ x(s) ~ \mathrm{d}{s};$$ the Hammerstein operator $$A(x) = \int_{a}^{b} K(t,s) ~ g(s,x(s)) ~ \mathrm{d}{s};$$ and the Urysohn (Uryson) operator $$A(x) = \int_{a}^{b} K(t,s,x(s)) ~ \mathrm{d}{s},$$ in certain function spaces, under suitable restrictions on the functions $K(t,s)$, $g(t,u)$ and $K(t,s,u)$.