Mackey-Arens' theorem

The Mackey-Arens theorem (after George Mackey and Richard Friederich Arens ) is a mathematical theorem from functional analysis , more precisely from the theory of locally convex spaces . Mackey-Arens' theorem deals with the question in which topologies certain important mappings are continuous .

More precisely, a locally convex space with a topology is given. Then one considers the dual space E 'of the functionals which are continuous and linear with respect to . The question now is which other locally convex topologies lead to the same continuous, linear functional as . Such topologies are called permissible. ${\displaystyle E}$ ${\displaystyle t}$ ${\displaystyle t}$ ${\displaystyle E}$ ${\displaystyle E}$ ${\displaystyle t}$

It turns out that there is a weakest and a strongest feasible topology.

The weakest allowable topology

The weakest allowable topology, i.e. H. the weakest topology, with respect to which all functionals from E 'are continuous, is the weak topology . It is clear that there cannot be a legal topology that is genuinely weaker, and it is not difficult to show that itself is legal. ${\displaystyle \sigma (E,E')}$ ${\displaystyle \sigma (E,E')}$

The Mackey topology

The dual space E 'carries the weak - * - topology , that is the weakest topology on E', which makes all mappings of the form , where , continuous. Let be the set of all absolutely convex and weak - * - compact sets . To be through defined seminorm on . Then the amount defined a locally convex topology on which one the Mackey topology on call and called. If one identifies with , d. H. with a function on E ', the Mackey topology is nothing other than the topology of uniform convergence on absolutely convex, weakly - * - compact sets. ${\displaystyle {\hat {x}}:E'\rightarrow {\mathbb {K} },\,f\mapsto f(x)}$ ${\displaystyle x\in E}$ ${\displaystyle {\mathcal {M}}}$ ${\displaystyle M\subset E'}$ ${\displaystyle M\in {\mathcal {M}}}$ ${\displaystyle p_{M}}$ ${\displaystyle \textstyle p_{M}(x):=\sup _{f\in M}|f(x)|}$ ${\displaystyle E}$ ${\displaystyle \{p_{M};M\in {\mathcal {M}}\}}$ ${\displaystyle E}$ ${\displaystyle E}$ ${\displaystyle \tau (E,E')}$ ${\displaystyle x}$ ${\displaystyle {\hat {x}}}$

It now shows that one can characterize the admissible topologies with the Mackey topology.

Mackey-Arens' theorem

• Is a locally convex space, then a locally convex topology on exactly permissible if . ${\displaystyle E}$ ${\displaystyle s}$ ${\displaystyle E}$ ${\displaystyle \sigma (E,E')\subset s\subset \tau (E,E')}$

According to this theorem, the Mackey topology is the strongest admissible topology ; the existence of such a topology is not obvious! By definition, the output topology of is itself permissible, i.e. it is also between and . If the output topology of corresponds to the Mackey topology, it is called a Mackey space . One can show that quasi-tiled rooms are always Mackey rooms. In particular, all barreled and all bornological spaces are therefore Mackey spaces. ${\displaystyle E}$ ${\displaystyle E}$ ${\displaystyle \sigma (E,E')}$ ${\displaystyle \tau (E,E')}$ ${\displaystyle E}$ ${\displaystyle E}$

Mackey's theorem

A lot of a locally convex space is limited , if for every zero neighborhood one is with . The restriction therefore depends on the topology. Hence the following theorem from Mackey is noteworthy: ${\displaystyle B}$ ${\displaystyle U}$ ${\displaystyle r>0}$ ${\displaystyle B\subset rU}$

For a subset of a locally convex space are equivalent: ${\displaystyle B}$

• ${\displaystyle B}$is limited in terms of topology to . ${\displaystyle E}$
• ${\displaystyle B}$ is restricted with regard to any permissible topology.
• ${\displaystyle B}$is limited in terms of . ${\displaystyle \sigma (E,E')}$
• ${\displaystyle B}$is limited in terms of . ${\displaystyle \tau (E,E')}$

meaning

The Mackey and Mackey-Arens theorems and Mackey topology play an important role in the duality theory of locally convex spaces. You will find u. a. Application in the characterization of semi-reflexivity . Further conclusions are theorems of Art

• The weak dual space of a barreled space is consequently complete .
• The weak dual space of a Fréchet space , which is not a Banach space , cannot be metrized.

In mathematical economics, so-called preference or utility functions appear on certain spaces in which the weak - * - topology of - duality is considered. These utility functions are generally discontinuous with respect to the weak - * - topology but continuous with respect to the finer Mackey topology . ${\displaystyle L^{\infty }}$ ${\displaystyle L^{1}}$ ${\displaystyle L^{\infty }}$ ${\displaystyle \tau (L^{\infty },L^{1})}$

literature

• K. Floret, J. Wloka: Introduction to the Theory of Locally Convex Spaces , Lecture Notes in Mathematics 56, 1968
• R. Meise, D. Vogt: Introduction to Functional Analysis , Vieweg, 1992 ISBN 3-528-07262-8