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.

An operator between two normed spaces is a bounded linear operator if and only if it is a continuous linear operator.

Continuous linear operators

See also: Continuous function (topology) and Discontinuous linear map

Characterizations of continuity

See also: Bounded operator

Suppose that $F:X\to Y$ is a linear operator between two topological vector spaces (TVSs). The following are equivalent:

1. $F$ is continuous.
2. $F$ is continuous at some point $x\in X.$
3. $F$ is continuous at the origin in $X.$

If $Y$ is locally convex then this list may be extended to include:

4. for every continuous seminorm $q$ on $Y,$ there exists a continuous seminorm $p$ on $X$ such that $q\circ F\leq p.$

If $X$ and $Y$ are both Hausdorff locally convex spaces then this list may be extended to include:

5. $F$ is weakly continuous and its transpose ${}^{t}F:Y^{\prime }\to X^{\prime }$ maps equicontinuous subsets of $Y^{\prime }$ to equicontinuous subsets of $X^{\prime }.$

If $X$ is a sequential space (such as a pseudometrizable space) then this list may be extended to include:

6. $F$ is sequentially continuous at some (or equivalently, at every) point of its domain.

If $X$ is pseudometrizable or metrizable (such as a normed or Banach space) then we may add to this list:

7. $F$ is a bounded linear operator (that is, it maps bounded subsets of $X$ to bounded subsets of $Y$).

If $Y$ is seminormable space (such as a normed space) then this list may be extended to include:

8. $F$ maps some neighborhood of 0 to a bounded subset of $Y.$

If $X$ and $Y$ are both normed or seminormed spaces (with both seminorms denoted by $\|\cdot \|$) then this list may be extended to include:

9. for every $r>0$ there exists some $\delta >0$ such that
   $${\text{ for all }}x,y\in X,{\text{ if }}\|x-y\|<\delta {\text{ then }}\|Fx-Fy\|<r.$$

If $X$ and $Y$ are Hausdorff locally convex spaces with $Y$ finite-dimensional then this list may be extended to include:

10. the graph of $F$ is closed in $X\times Y.$

Continuity and boundedness

Throughout, $F:X\to Y$ is a linear map between topological vector spaces (TVSs).

Bounded on a set

See also: Bounded set (topological vector space)

The notion of "bounded set" for a topological vector space is that of being a von Neumann bounded set. If the space happens to also be a normed space (or a seminormed space), such as the scalar field with the absolute value for instance, then a subset $S$ is von Neumann bounded if and only if it is norm bounded; that is, if and only if $\sup _{s\in S}\|s\|<\infty .$ If $S\subseteq X$ is a set then $F:X\to Y$ is said to be bounded on $S$ if $F(S)$ is a bounded subset of $Y,$ which if $(Y,\|\cdot \|)$ is a normed (or seminormed) space happens if and only if $\sup _{s\in S}\|F(s)\|<\infty .$ A linear map $F$ is bounded on a set $S$ if and only if it is bounded on $x+S$ for every $x\in X$ (because $F(x+S)=F(x)+F(S)$ and any translation of a bounded set is again bounded).

Bounded linear maps

See also: Bounded linear operator

By definition, a linear map $F:X\to Y$ between TVSs is said to be bounded and is called a bounded linear operator if for every (von Neumann) bounded subset $B\subseteq X$ of its domain, $F(B$