# Standardizable space

A normalizable space or normalizable vector space is a topological vector space in mathematics , the topology of which can be generated by a norm . Normable spaces are examined in particular in topology and functional analysis.

## definition

A topological vector space is called normalizable if there is a norm for such that the system of sets with ${\ displaystyle (V, {\ mathcal {T}})}$ ${\ displaystyle \ | \ cdot \ |}$${\ displaystyle V}$${\ displaystyle (U _ {\ varepsilon}) _ {\ varepsilon> 0}}$

${\ displaystyle U _ {\ varepsilon}: = \ {v \ in V: \ | v \ | \ leq \ varepsilon \}}$

form an environment base of the zero vector with respect to the topology . This is equivalent to the topology being induced on by the standard . ${\ displaystyle {\ mathcal {T}}}$${\ displaystyle V}$${\ displaystyle \ | \ cdot \ |}$

## properties

In general, the topology of a normalizable space can be generated by several norms. If and are two norms that produce the same topology, then these two norms are equivalent to one another . If one of the possible norms is selected, then a standardized space becomes the norm topology of which corresponds. ${\ displaystyle \ | \ cdot \ | _ {a}}$${\ displaystyle \ | \ cdot \ | _ {b}}$${\ displaystyle V}$${\ displaystyle {\ mathcal {T}}}$

Scalability remains under the following operations:

## Criteria for normalization

According to Kolmogoroff's criterion for normalization, a Hausdorff topological vector space can be normalized if and only if it has a bounded and convex null neighborhood. In particular, every Hausdorff locally convex space with a bounded zero neighborhood can be normalized.

Examples of non normalizable topological vector spaces are all not locally convex spaces, in particular L p ([0,1]) where 0 < p <1 , and all the infinite- Montel spaces , in particular spaces , , , , and the distribution theory . Further examples for non-normalizable topological vector spaces are provided by the weak topology on infinite-dimensional normalized spaces , because the space can be normalized if and only if is finite-dimensional. ${\ displaystyle {\ mathcal {D}} \ left (\ Omega \ right)}$${\ displaystyle {\ mathcal {S}} \ left (\ Omega \ right)}$${\ displaystyle {\ mathcal {E}} \ left (\ Omega \ right)}$${\ displaystyle {\ mathcal {E}} '\ left (\ Omega \ right)}$${\ displaystyle {\ mathcal {S}} '\ left (\ Omega \ right)}$${\ displaystyle {\ mathcal {D}} '\ left (\ Omega \ right)}$ ${\ displaystyle \ sigma}$${\ displaystyle E}$${\ displaystyle \ left (E, \ sigma \ right)}$${\ displaystyle E}$

## literature

• Jürgen Heine: Topology and Functional Analysis: Basics of Abstract Analysis with Applications . 2nd Edition. de Gruyter, 2012, ISBN 978-3-486-71968-0 .
• John Leroy Kelley, Isaac Namioka: Linear Topological Spaces . Springer, 2013, ISBN 978-3-662-41914-4 .
• Helmut H. Schaefer: Topological Vector Spaces (=  Graduate Texts in Mathematics . Volume 3 ). Springer, 2013, ISBN 978-1-4684-9928-5 .