Content deleted Content added
Line 85:
For any linear map, if it is [[#bounded on a neighborhood|bounded on a neighborhood]] then it is continuous,{{sfn|Narici|Beckenstein|2011|pp=156-175}}{{sfn|Wilansky|2013|pp=54-55}} and if it is continuous then it is [[Bounded linear operator|bounded]].{{sfn|Narici|Beckenstein|2011|pp=441-457}} The converse statements are not true in general but they are both true when the linear map's ___domain is a [[normed space]]. Examples and additional details are now given below.
====Continuous and bounded but not bounded on a neighborhood====
The next example shows that it is possible for a linear map to be [[Continuous function (topology)|continuous]] (and thus also bounded) but not bounded on any neighborhood. In particular, it demonstrates that being "bounded on a neighborhood" is {{em|not}} always synonymous with being "[[Bounded linear operator|bounded]]".
Line 92:
This shows that it is possible for a linear map to be continuous but {{em|not}} bounded on any neighborhood.
Indeed, this example shows that every [[Locally convex topological vector space|locally convex space]] that is not seminormable has a linear TVS-[[automorphism]] that is not bounded on any neighborhood of any point.
Thus although every linear map that is bounded on a neighborhood is necessarily continuous, the converse is not guaranteed in general.
===Guaranteeing converses===
| |||