Ultrabornological room

Ultrabornological spaces are examined in the mathematical sub-area of functional analysis. These are bornological spaces with a certain additional completeness property. These spaces get their meaning from the fact that they can be used to prove generalizations of two central theorems from the theory of Banach spaces , namely the theorem about the open mapping and the theorem about the closed graph .

Motivation and Definition

If a locally convex space and a bounded and absolutely convex subset is a vector space and the Minkowski functional of makes this vector space a normalized space . If this normalized space is even a Banach space , it is called a Banach sphere . ${\displaystyle E}$ ${\displaystyle B}$ ${\displaystyle E_{B}:=\bigcup _{\lambda >0}\lambda \cdot B}$ ${\displaystyle p_{B}}$ ${\displaystyle B\subset E_{B}}$ ${\displaystyle B}$

A characterization of the bornological spaces reads: A locally convex space is bornological if and only if the continuity of a linear mapping in any other locally convex space already results from the fact that the image of every limited set is limited. Therefore, the following definition represents a tightening of this property: ${\displaystyle E}$

A locally convex space is called ultrabornological if every linear mapping from into another locally convex space is already continuous if the image of every Banach sphere is bounded. ${\displaystyle E}$ ${\displaystyle E}$

properties

• As stated above, the definition immediately shows that ultrabornological spaces are bornological.
• Ultrabornological spaces are barreled , which is generally wrong for bornological spaces.
• An ultrabornological space has the finest locally convex topology for which all embeddings are continuous, with all Banach spheres running through. In this sense, ultrabornological spaces have an additional completeness property compared to the bornological spaces. ${\displaystyle E_{B}\subset E}$ ${\displaystyle B}$
• Inductive limits of ultrabornological spaces are again ultrabornological.

Examples

• The ultrabornological rooms are grouped into other classes of rooms as follows, whereby many examples are given at the same time. Sequence completeness means that every Cauchy sequence converges.

Banach room Fréchet room (LF) room follow complete bornological room ultrabornological room ${\displaystyle \Rightarrow }$ ${\displaystyle \Rightarrow }$ ${\displaystyle \Rightarrow }$ ${\displaystyle \Rightarrow }$

• Let be a compact space and the vector space of continuous functions with the strict topology , i.e. H. with the topology given by the semi-norms , with the restricted functions defined on . Then this space is ultrabornological. ${\displaystyle X}$ ${\displaystyle C(X)}$ ${\displaystyle X\rightarrow {\mathbb {C} }}$ ${\displaystyle p_{\phi }(f):=\sup _{x\in X}|f(x)\phi (x)|}$ ${\displaystyle \phi }$ ${\displaystyle X}$

• The strong dual space of a complete black space is ultrabornological.

Graph set and openness

General versions of the theorem about the open mapping and the theorem about the closed graph result in interaction with spaces with tissue , Fréchet spaces are examples of such spaces.

Theorem about the open mapping : Be a space with tissue, be ultrabornological and be linear, continuous and surjective. Then it's open. ${\displaystyle E}$ ${\displaystyle F}$ ${\displaystyle T:E\rightarrow F}$ ${\displaystyle T}$

Theorem of the closed graph : Be ultrabornological, be a space with tissue, be a linear operator with a closed graph. Then is steady. ${\displaystyle E}$ ${\displaystyle F}$ ${\displaystyle T:E\rightarrow F}$ ${\displaystyle T}$

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

swell

• H. Jarchow: Locally Convex Spaces , Teubner, Stuttgart 1981 ISBN 3-519-02224-9
• Reinhold Meise, Dietmar Vogt: Introduction to functional analysis (= Vieweg study 62 advanced course in mathematics ). Vieweg, Braunschweig et al. 1992, ISBN 3-528-07262-8 .