Editing Bounded operator (section)

===Continuity and boundedness===

Every [[sequentially continuous]] linear operator between TVS is a bounded operator.{{sfn|Wilansky|2013|pp=47-50}} 
This implies that every continuous linear operator between metrizable TVS is bounded. 
However, in general, a bounded linear operator between two TVSs need not be continuous.

This formulation allows one to define bounded operators between general topological vector spaces as an operator which takes bounded sets to bounded sets. 
In this context, it is still true that every continuous map is bounded, however the converse fails; a bounded operator need not be continuous. 
This also means that boundedness is no longer equivalent to Lipschitz continuity in this context.

If the domain is a [[bornological space]] (for example, a [[Metrizable topological vector space|pseudometrizable TVS]], a [[Fréchet space]], a [[normed space]]) then a linear operators into any other locally convex spaces is bounded if and only if it is continuous. 
For [[LF space]]s, a weaker converse holds; any bounded linear map from an LF space is [[sequentially continuous]].

If <math>F : X \to Y</math> is a linear operator between two topological vector spaces and if there exists a neighborhood <math>U</math> of the origin in <math>X</math> such that <math>F(U)</math> is a bounded subset of <math>Y,</math> then <math>F</math> is continuous.{{sfn|Narici|Beckenstein|2011|pp=156-175}} 
This fact is often summarized by saying that a linear operator that is bounded on some neighborhood of the origin is necessarily continuous. 
In particular, any linear functional that is bounded on some neighborhood of the origin is continuous (even if its domain is not a [[normed space]]). 

====Bornological spaces====
{{Main|Bornological space}}

Bornological spaces are exactly those locally convex spaces for which every bounded linear operator into another locally convex space is necessarily continuous. 
That is, a locally convex TVS <math>X</math> is a bornological space if and only if for every locally convex TVS <math>Y,</math> a linear operator <math>F : X \to Y</math> is continuous if and only if it is bounded.{{sfn|Narici|Beckenstein|2011|pp=441-457}}

Every normed space is bornological.