Namespaces
Variants
Actions

Difference between revisions of "Mackey topology"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
(TeX)
Line 1: Line 1:
''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062080/m0620801.png" /> on a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062080/m0620802.png" />, being in duality with a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062080/m0620803.png" /> (over the same field)''
+
{{TEX|done}}
 +
''$\tau(F,G)$ on a space $F$, being in duality with a space $G$ (over the same field)''
  
The topology of uniform convergence on the convex balanced subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062080/m0620804.png" /> that are compact in the weak topology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062080/m0620805.png" /> (defined by the duality between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062080/m0620806.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062080/m0620807.png" />). It was introduced by G.W. Mackey [[#References|[1]]]. The Mackey topology is the strongest of the separated locally convex topologies (cf. [[Locally convex topology|Locally convex topology]]) which are compatible with the duality between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062080/m0620808.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062080/m0620809.png" /> (that is, separated locally convex topologies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062080/m06208010.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062080/m06208011.png" /> such that the set of all continuous linear functionals on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062080/m06208012.png" /> endowed with the topology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062080/m06208013.png" /> coincides with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062080/m06208014.png" />). The families of sets in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062080/m06208015.png" /> which are bounded relative to the Mackey topology and bounded relative to the weak topology coincide. A convex subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062080/m06208016.png" /> is equicontinuous when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062080/m06208017.png" /> is endowed with the Mackey topology if and only if it is relatively compact in the weak topology. If a separated [[Locally convex space|locally convex space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062080/m06208018.png" /> is barrelled or bornological (in particular, metrizable) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062080/m06208019.png" /> is its dual, then the Mackey topology on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062080/m06208020.png" /> (being dual with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062080/m06208021.png" />) coincides with the initial topology on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062080/m06208022.png" />. For pairs of spaces (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062080/m06208023.png" />) in duality the Mackey topology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062080/m06208024.png" /> is not necessarily barrelled or metrizable. A weakly-continuous linear mapping of a separated locally convex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062080/m06208025.png" /> into a separated locally convex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062080/m06208026.png" /> is continuous relative to the Mackey topologies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062080/m06208027.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062080/m06208028.png" />. A locally convex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062080/m06208029.png" /> is called a Mackey space if the topology on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062080/m06208030.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062080/m06208031.png" />. Completions, quotient spaces and metrizable subspaces, products, locally convex direct sums, and inductive limits of families of Mackey spaces are Mackey spaces. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062080/m06208032.png" /> is a Mackey space and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062080/m06208033.png" /> is a weakly-continuous mapping of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062080/m06208034.png" /> into a locally convex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062080/m06208035.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062080/m06208036.png" /> is a continuous linear mapping of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062080/m06208037.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062080/m06208038.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062080/m06208039.png" /> is a quasi-complete Mackey space and the space dual to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062080/m06208040.png" /> equipped with the strong <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062080/m06208041.png" />-topology is semi-reflexive, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062080/m06208042.png" /> is reflexive.
+
The topology of uniform convergence on the convex balanced subsets of $G$ that are compact in the weak topology $\sigma(F,G)$ (defined by the duality between $F$ and $G$). It was introduced by G.W. Mackey [[#References|[1]]]. The Mackey topology is the strongest of the separated locally convex topologies (cf. [[Locally convex topology|Locally convex topology]]) which are compatible with the duality between $F$ and $G$ (that is, separated locally convex topologies $\mathcal T$ on $F$ such that the set of all continuous linear functionals on $F$ endowed with the topology $\mathcal T$ coincides with $G$). The families of sets in $F$ which are bounded relative to the Mackey topology and bounded relative to the weak topology coincide. A convex subsets of $G$ is equicontinuous when $F$ is endowed with the Mackey topology if and only if it is relatively compact in the weak topology. If a separated [[Locally convex space|locally convex space]] $E$ is barrelled or bornological (in particular, metrizable) and $E'$ is its dual, then the Mackey topology on $E$ (being dual with $E'$) coincides with the initial topology on $E$. For pairs of spaces ($F,G$) in duality the Mackey topology $\mathcal T$ is not necessarily barrelled or metrizable. A weakly-continuous linear mapping of a separated locally convex space $E$ into a separated locally convex space $F$ is continuous relative to the Mackey topologies $\tau(E,E')$ and $\tau(F,F')$. A locally convex space $E$ is called a Mackey space if the topology on $E$ is $\tau(E,E')$. Completions, quotient spaces and metrizable subspaces, products, locally convex direct sums, and inductive limits of families of Mackey spaces are Mackey spaces. If $E$ is a Mackey space and $\phi$ is a weakly-continuous mapping of $E$ into a locally convex space $F$, then $\phi$ is a continuous linear mapping of $E$ into $F$. If $E$ is a quasi-complete Mackey space and the space dual to $E$ equipped with the strong $E$-topology is semi-reflexive, then $E$ is reflexive.
  
 
====References====
 
====References====

Revision as of 15:39, 17 July 2014

$\tau(F,G)$ on a space $F$, being in duality with a space $G$ (over the same field)

The topology of uniform convergence on the convex balanced subsets of $G$ that are compact in the weak topology $\sigma(F,G)$ (defined by the duality between $F$ and $G$). It was introduced by G.W. Mackey [1]. The Mackey topology is the strongest of the separated locally convex topologies (cf. Locally convex topology) which are compatible with the duality between $F$ and $G$ (that is, separated locally convex topologies $\mathcal T$ on $F$ such that the set of all continuous linear functionals on $F$ endowed with the topology $\mathcal T$ coincides with $G$). The families of sets in $F$ which are bounded relative to the Mackey topology and bounded relative to the weak topology coincide. A convex subsets of $G$ is equicontinuous when $F$ is endowed with the Mackey topology if and only if it is relatively compact in the weak topology. If a separated locally convex space $E$ is barrelled or bornological (in particular, metrizable) and $E'$ is its dual, then the Mackey topology on $E$ (being dual with $E'$) coincides with the initial topology on $E$. For pairs of spaces ($F,G$) in duality the Mackey topology $\mathcal T$ is not necessarily barrelled or metrizable. A weakly-continuous linear mapping of a separated locally convex space $E$ into a separated locally convex space $F$ is continuous relative to the Mackey topologies $\tau(E,E')$ and $\tau(F,F')$. A locally convex space $E$ is called a Mackey space if the topology on $E$ is $\tau(E,E')$. Completions, quotient spaces and metrizable subspaces, products, locally convex direct sums, and inductive limits of families of Mackey spaces are Mackey spaces. If $E$ is a Mackey space and $\phi$ is a weakly-continuous mapping of $E$ into a locally convex space $F$, then $\phi$ is a continuous linear mapping of $E$ into $F$. If $E$ is a quasi-complete Mackey space and the space dual to $E$ equipped with the strong $E$-topology is semi-reflexive, then $E$ is reflexive.

References

[1] G.W. Mackey, "On convex topological linear spaces" Trans. Amer. Math. Soc. , 60 (1946) pp. 519–537
[2] N. Bourbaki, "Elements of mathematics. Topological vector spaces" , Addison-Wesley (1977) (Translated from French)
[3] H.H. Schaefer, "Topological vector spaces" , Springer (1971)


Comments

References

[a1] H. Jarchow, "Locally convex spaces" , Teubner (1981) (Translated from German)
How to Cite This Entry:
Mackey topology. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Mackey_topology&oldid=13416
This article was adapted from an original article by A.I. Shtern (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article