Mackey topology

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

More specifically, a locally convex space is given with a topology. Then one considers the dual space E ' with respect to the continuous linear functionals. The question now is, what other locally convex topologies on the same continuous linear functionals such as lead. Such topologies are called admissible.

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

The weakest admissible topology

The weakest admissible topology, that is, the weakest topology with respect to which all the functionals E ' are continuous, the weak topology. It is clear that there can be no admissible topology that is strictly slower, and it is not difficult to show that even is allowed.

The Mackey topology

The dual space E ' bears the weak -* topology, which is the weakest topology on E' which makes all the pictures of the mold, constantly. Let the set of all absolutely convex and weak - * - compact sets. To be made to the semi-norm defined by. Then define the amount on a locally convex topology, which one calls on the Mackey topology and is denoted by. If we identify with, that is, with a function on E ', then the Mackey topology nothing more than the topology of uniform convergence on absolutely convex, weakly -* compact sets.

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

Set of Mackey - Arens

  • If a locally convex space, then a locally convex topology on exactly permissible if.

After this block, the Mackey topology is the strongest admissible topology, the existence of such a topology is not obvious! The output topology is by definition self- admissible, so also lies between and. Is the output topology consistent with the Mackey topology, it is called a Mackey space. One can show that quasi- barreled spaces are always Mackey spaces. In particular, therefore all are barreled and bornological spaces Mackey all rooms.

Set of Mackey

A of a locally convex space is called bounded if for every zero neighborhood is a with. The boundedness therefore depends on the topology. Therefore, the following set of Mackey is remarkable:

For a subset of a locally convex space are equivalent:

  • Is limited with respect to the topology.
  • Is limited with respect to each admissible topology.
  • Is limited with respect.
  • Is limited with respect.

Importance

The sets of Mackey and Mackey - Arens and the Mackey topology play an important role in the duality theory of locally convex spaces. See, inter alia, Application in the characterization of the semi-reflexivity. Other inferences are sets of type

  • The weak dual space of a barreled space is sequentially complete.
  • The weak dual space of a Fréchet space which is not a Banach space is not metrizable.

In mathematical economics occur so-called preference function or utility functions on certain spaces on which one the weak -* topology of - considered duality. These utility functions are in general discontinuous with respect to the weak -* topology but continuous with respect to the finer Mackey topology.

Mathematics Functional (mathematics)
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