Webbed space

Rooms with tissues should be considered in the mathematical discipline of functional analysis. They allow the interaction with the ultrabornologischen rooms generalizations of two central theorems from the theory of Banach spaces, which are set via the open mapping and closed graph from the set. These spaces were introduced in 1969 by Marc de Wilde for exactly this purpose.

The definition is very technical, but it can be apart from the special technical conditions in many applications because it can be shown that large classes of topological vector spaces have this property, and that, therefore, the generalizations of the above statements, and these are in the applications essential.

Rooms with tissue can be defined for arbitrary topological vector spaces. Are considered here for the sake of simplicity only locally convex spaces. The general theory of topological vector spaces is discussed in the below textbook by H. Jarchow.

Tissue

A fabric in a locally convex space is a family of subsets, so that the following applies:

One can imagine the amounts as a finer and finer with increasing nascent web that spans the room, imagine what explains the name of the tissue.

Is there such a fabric in a locally convex space, then we say the room had a tissue or had a room with a tissue. The German term sounds a bit wooden, the English name webbed space can not be in the German play so rough.

Permanenzeigenschaften

Rooms with fabrics have very extensive Permanenzeigenschaften:

  • If a room with fabric and a closed subspace, so are the quotient space and rooms with tissue.
  • Is a sequence of locally convex areas of tissue, the direct product of the topology is a product chamber with tissue.
  • Is a sequence of locally convex areas of tissue, the direct sum of the final topology is a space with tissue.

Examples

  • Banach spaces have a tissue. Namely, the ball unit, thus forming the data (regardless of the sequence! ) A fabric.
  • Since every Fréchet space is a closed subspace of a countable direct product of Banach spaces, resulting from the above Permanenzeigenschaften that Fréchet spaces have a tissue.
  • Similarly, it follows from the above Permanenzeigenschaften that countable inductive limits of Fréchet spaces have a tissue, because these occur as the ratio of direct sums of countable Fréchet spaces. In particular, LF rooms have a tissue.
  • Full episode (DF )-spaces are spaces with tissue.

Graph set and openness

For linear operators between spaces with tissue and ultrabornologischen spaces can prove the theorem by the closed graph and the set of the open mapping.

Set over the open picture: Be a room with fabric, be ultrabornologisch and is linear, continuous and surjective. Then is open.

Set by the closed graph: Be ultrabornologisch, is a room with fabric, is a linear operator with closed graph. Then is continuous.

Note the changing roles of spatial classes in these two sentences, (LF )-spaces belong to both classes.

Swell

  • G. Köthe: Topological Vector Spaces II, Springer, 1979, ISBN 3-540-90400- X
  • H. Jarchow: Locally Convex Spaces, Teubner, Stuttgart 1981 ISBN 3-519-02224-9
  • R. Meise, D. Vogt: Introduction to Functional Analysis, Vieweg, 1992 ISBN 3-528-07262-8
  • Topological vector space
  • Functional Analysis
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