Continuous linear operator

In functional analysis and related areas of mathematics, a continuous linear operator or continuous linear mapping is a continuous linear transformation between topological vector spaces. In functional analysis and related areas of mathematics, a continuous linear operator or continuous linear mapping is a continuous linear transformation between topological vector spaces. An operator between two normed spaces is a bounded linear operator if and only if it is a continuous linear operator. If for every ϵ > 0 {displaystyle epsilon >0} there exists a δ > 0 {displaystyle delta >0} such that | | x − y | | < δ ⇒ | | A x − A y | | < ϵ {displaystyle ||x-y||<delta Rightarrow ||Ax-Ay||<epsilon } we say the operator A {displaystyle A} is continuous. A continuous linear operator maps bounded sets into bounded sets. A linear functional is continuous if and only if its kernel is closed. Every linear function on a finite-dimensional space is continuous. The following are equivalent: given a linear operator A between topological spaces X and Y: