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

Characterizations of continuity

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

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

If ${\displaystyle Y}$ is locally convex then this list may be extended to include:

4. for every continuous seminorm ${\displaystyle q}$ on ${\displaystyle Y,}$ there exists a continuous seminorm ${\displaystyle p}$ on ${\displaystyle X}$ such that ${\displaystyle q\circ F\leq p.}$^[1]

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

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

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

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

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

7. ${\displaystyle F}$ is a bounded linear operator (that is, it maps bounded subsets of ${\displaystyle X}$ to bounded subsets of ${\displaystyle Y}$).^[2]

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

8. ${\displaystyle F}$ maps some neighborhood of 0 to a bounded subset of ${\displaystyle Y.}$^[3]

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

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

   

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

10. the graph of ${\displaystyle F}$ is closed in ${\displaystyle X\times Y.}$^[4]

Continuity and boundedness

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

Bounded subset

The notion of a "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) then a subset ${\displaystyle S}$ is von Neumann bounded if and only if it is norm bounded, meaning that ${\displaystyle \sup _{s\in S}\|s\|<\infty .}$ A subset of a normed (or seminormed) space is called bounded if it is norm-bounded (or equivalently, von Neumann bounded). For example, the scalar field (${\displaystyle \mathbb {R} }$ or ${\displaystyle \mathbb {C} }$) with the absolute value ${\displaystyle |\cdot |}$ is a normed space, so a subset ${\displaystyle S}$ is bounded if and only if ${\displaystyle \sup _{s\in S}|s|}$ is finite, which happens if and only if ${\displaystyle S}$ is contained in some open (or closed) ball centered at the origin (zero).

Any translation, scalar multiple, and subset of a bounded set is again bounded.

Function bounded on a set

If ${\displaystyle S\subseteq X}$ is a set then ${\displaystyle F:X\to Y}$ is said to be bounded on ${\displaystyle S}$ if ${\displaystyle F(S)}$ is a bounded subset of ${\displaystyle Y,}$ which if ${\displaystyle (Y,\|\cdot \|)}$ is a normed (or seminormed) space happens if and only if ${\displaystyle \sup _{s\in S}\|F(s)\|<\infty .}$ A linear map ${\displaystyle F}$ is bounded on a set ${\displaystyle S}$ if and only if it is bounded on ${\displaystyle x+S:=\{x+s:s\in S\}}$ for every ${\displaystyle x\in <!--\; \in \; -->X}$Zdroj:https://en.wikipedia.org?pojem=Continuous_linear_operator
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