Editing Normed vector space (section)

== Normable spaces ==

{{See also|Metrizable topological vector space#Normability}}

A [[topological vector space]] <math>(X, \tau)</math> is called '''normable''' if there exists a norm <math>\| \cdot \|</math> on <math>X</math> such that the canonical metric <math>(x, y) \mapsto \|y-x\|</math> induces the topology <math>\tau</math> on <math>X.</math>
The following theorem is due to [[Andrey Kolmogorov|Kolmogorov]]:{{sfn|Schaefer|1999|p=41}}

'''[[Kolmogorov's normability criterion]]''': A Hausdorff topological vector space is normable if and only if there exists a convex, [[von Neumann bounded]] neighborhood of <math>0 \in X.</math>

A product of a family of normable spaces is normable if and only if only finitely many of the spaces are non-trivial (that is, <math>\neq \{ 0 \}</math>).{{sfn|Schaefer|1999|p=41}} Furthermore, the quotient of a normable space <math>X</math> by a closed vector subspace <math>C</math> is normable, and if in addition <math>X</math>'s topology is given by a norm <math>\|\,\cdot,\|</math> then the map <math>X/C \to \R</math> given by <math display=inline>x + C \mapsto \inf_{c \in C} \|x + c\|</math> is a well defined norm on <math>X / C</math> that induces the [[quotient topology]] on <math>X / C.</math>{{sfn|Schaefer|1999|p=42}}

If <math>X</math> is a Hausdorff [[Locally convex topological vector space|locally convex]] [[topological vector space]] then the following are equivalent:

# <math>X</math> is normable.
# <math>X</math> has a bounded neighborhood of the origin.
# the [[strong dual space]] <math>X^{\prime}_b</math> of <math>X</math> is normable.{{sfn|Trèves|2006|pp=136–149, 195–201, 240–252, 335–390, 420–433}}
# the strong dual space <math>X^{\prime}_b</math> of <math>X</math> is [[Metrizable topological vector space|metrizable]].{{sfn|Trèves|2006|pp=136–149, 195–201, 240–252, 335–390, 420–433}}

Furthermore, <math>X</math> is finite-dimensional if and only if <math>X^{\prime}_{\sigma}</math> is normable (here <math>X^{\prime}_{\sigma}</math> denotes <math>X^{\prime}</math> endowed with the [[weak-* topology]]).

The topology <math>\tau</math> of the [[Fréchet space]] <math>C^{\infty}(K),</math> as defined in the article on [[spaces of test functions and distributions]], is defined by a countable family of norms but it is {{em|not}} a normable space because there does not exist any norm <math>\|\cdot\|</math> on <math>C^{\infty}(K)</math> such that the topology that this norm induces is equal to <math>\tau.</math> 

Even if a metrizable topological vector space has a topology that is defined by a family of norms, then it may nevertheless still fail to be [[normable space]] (meaning that its topology can not be defined by any {{em|single}} norm). 
An example of such a space is the [[Fréchet space]] <math>C^{\infty}(K),</math> whose definition can be found in the article on [[spaces of test functions and distributions]], because its topology <math>\tau</math> is defined by a countable family of norms but it is {{em|not}} a normable space because there does not exist any norm <math>\|\cdot\|</math> on <math>C^{\infty}(K)</math> such that the topology this norm induces is equal to <math>\tau.</math>  
In fact, the topology of a [[Locally convex topological vector space|locally convex space]] <math>X</math> can be a defined by a family of {{em|norms}} on <math>X</math> if and only if there exists {{em|at least one}} continuous norm on <math>X.</math>{{sfn|Jarchow|1981|p=130}}