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:

• Every subspace of a normalizable space can be normalized again.
• The factor space of a normalizable space after a closed sub-vector space can be normalized again.
• The direct product of a family of normalizable spaces can be normalized again if only finitely many of these spaces are not equal to the zero vector space .
• The completion of a normalizable space can be normalized again.

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. ${\displaystyle {\mathcal {D}}\left(\Omega \right)}$ ${\displaystyle {\mathcal {S}}\left(\Omega \right)}$ ${\displaystyle {\mathcal {E}}\left(\Omega \right)}$ ${\displaystyle {\mathcal {E}}'\left(\Omega \right)}$ ${\displaystyle {\mathcal {S}}'\left(\Omega \right)}$ ${\displaystyle {\mathcal {D}}'\left(\Omega \right)}$ ${\displaystyle \sigma }$ ${\displaystyle E}$ ${\displaystyle \left(E,\sigma \right)}$ ${\displaystyle E}$

See also

• Quasi-normalizable space
• Metrizable space

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 .

Web links

• Norm . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).

Individual evidence

1. ↑ Jürgen Heine: Topology and Functional Analysis: Fundamentals of Abstract Analysis with Applications . 2nd Edition. de Gruyter, 2012, p. 35 .
2. ↑ John Leroy Kelley, Isaac Namioka: Linear Topological Spaces . Springer, 2013, p. 43 .
3. ↑ ^{a b c} Helmut H. Schaefer: Topological Vector Spaces (=  Graduate Texts in Mathematics . Volume 3 ). Springer, 2013, p. 41 .
4. ↑ Helmut H. Schaefer: Topological Vector Spaces (=  Graduate Texts in Mathematics . Volume 3 ). Springer, 2013, p. 42 .