Continuous linear operator
Encyclopedia
In functional analysis
and related areas of mathematics
, a continuous linear operator or continuous linear mapping is a continuous linear transformation
between topological vector space
s.
An operator between two normed spaces is a bounded linear operator if and only if it is a continuous linear operator.
s 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:
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.
Functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
and related areas of mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a continuous linear operator or continuous linear mapping is a continuous linear transformation
Linear transformation
In mathematics, a linear map, linear mapping, linear transformation, or linear operator is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. As a result, it always maps straight lines to straight lines or 0...
between topological vector space
Topological vector space
In mathematics, a topological vector space is one of the basic structures investigated in functional analysis...
s.
An operator between two normed spaces is a bounded linear operator if and only if it is a continuous linear operator.
Properties
A continuous linear operator maps bounded setBounded set (topological vector space)
In functional analysis and related areas of mathematics, a set in a topological vector space is called bounded or von Neumann bounded, if every neighborhood of the zero vector can be inflated to include the set...
s 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:
- A is continuous at 0 in X.
- A is continuous at some point
in X. - 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.