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Every linear map whose ___domain is a finite-dimensional Hausdorff [[topological vector space]] (TVS) is continuous. This is not true if the finite-dimensional TVS is not Hausdorff.
Every (constant) map <math>X \to Y</math> between TVS that is identically equal to zero is a linear map that is continuous, bounded, and bounded on the neighborhood <math>X</math> of the origin. In particular, every TVS has a non-empty [[continuous dual space]] (although it is possible for the constant zero map to be its only continuous linear functional).
Suppose <math>X</math> is any Hausdorff TVS. Then {{em|every}} [[linear functional]] on <math>X</math> is necessarily continuous if and only if every vector subspace of <math>X</math> is closed.{{sfn|Wilansky|2013|p=55}} Every linear functional on <math>X</math> is necessarily a bounded linear functional if and only if every [[Bounded set (topological vector space)|bounded subset]] of <math>X</math> is contained in a finite-dimensional vector subspace.{{sfn|Wilansky|2013|p=50}} ▼
▲Suppose <math>X</math> is any Hausdorff TVS. Then {{em|every}} [[linear functional]] on <math>X</math> is necessarily continuous if and only if every vector subspace of <math>X</math> is closed.{{sfn|Wilansky|2013|p=55}} Every linear functional on <math>X</math> is necessarily a bounded linear functional if and only if every [[Bounded set (topological vector space)|bounded subset]] of <math>X</math> is contained in a finite-dimensional vector subspace.{{sfn|Wilansky|2013|p=50}}
==Properties==
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