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. Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of spaces of functions. ... For other meanings of mathematics or math, see mathematics (disambiguation). ... In topology and related areas of mathematics a continuous function is a morphism between topological spaces. ... In mathematics, a linear transformation (also called linear map or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ... In mathematics a topological vector space is one of the basic structures investigated in functional analysis. ...

If such an operator is between two normed spaces, then it is a bounded linear operator (if fact this condition is also necessary). In mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the length of a vector is intuitive and can be easily extended to any real vector space Rn. ... In mathematics, the operator norm is a norm defined on the space of bounded operators between two Banach spaces. ...

Properties

A continuous linear operator maps bounded sets into bounded sets. In mathematical analysis and related areas of mathematics, a set is called bounded, if it is, in a certain sense, of finite size. ...

The following are equivalent: given a linear operator A between topological spaces X and Y:

(1) A is continuous at 0 in X.

(2) A is continuous at some point x₀ in X.

(3) A is continuous everywhere in X.

The proof uses the facts that the translation of an open set in a linear topological space is again an open set, and the equality

for any set D in Y and any x₀ in X, which is true due to the additivity of A.