Bornological space

Bornological spaces are special locally convex spaces in the mathematical subfield of functional analysis , for whose linear operators the equivalence of continuity and boundedness , known from the theory of normalized spaces, applies. These spaces can be characterized by their null environment bases and have other properties in common with normalized spaces.

motivation

A subset A of a topological K-vector space E is called bounded if it is absorbed by every null neighborhood, i.e. H. for every zero neighborhood there is a with . ${\displaystyle U\subset E}$ ${\displaystyle \lambda \in K}$ ${\displaystyle A\subset \lambda U}$

A subset B of a locally convex K-vector space is called a Bornolog if the following conditions are met:

• B is absolutely convex , i.e. H. for and with applies . ${\displaystyle x,y\in B}$ ${\displaystyle \lambda ,\mu \in K}$ ${\displaystyle |\lambda |+|\mu |\leq 1}$ ${\displaystyle \lambda x+\mu y\in B}$
• B absorbs any limited amount, i.e. H. for every limited amount there is one with . ${\displaystyle A\subset E}$ ${\displaystyle \lambda \in K}$ ${\displaystyle A\subset \lambda B}$

It is easy to show that every locally convex space has a zero neighborhood basis from Bornologists. Conversely, if every Bornolog is a null environment, the space is called bornological .

Examples

• Every metrizable locally convex space E is bornological. If B is a Bornolog in E, a countable zero neighborhood basis of E, and if one assumes that B does not contain a set of the form , one can choose one. Then converges , i.e. H. is compact and therefore restricted, that is, contained in a quantity of the form . For follows the contradiction . So B is a null neighborhood. ${\displaystyle U_{1}\supset U_{2}\supset \ldots }$ ${\displaystyle {\tfrac {1}{n}}U_{n}}$ ${\displaystyle x_{n}\in {\tfrac {1}{n}}U_{n}\setminus B}$ ${\displaystyle nx_{n}\rightarrow 0}$ ${\displaystyle \{nx_{n};n\in {\mathbb {N} }\}}$ ${\displaystyle \lambda B}$ ${\displaystyle n\geq \lambda }$ ${\displaystyle x_{n}\in B}$

• If E is a normalized space not equal to {0}, the final topology is an example of a bornological space that cannot be metrized. ${\displaystyle \textstyle \bigoplus _{r\in {\mathbb {R} }}E}$

Inheritance properties

An inductive limit of bornological spaces is again bornological.

Limited operators

As in the theory of normalized spaces, a linear operator between topological vector spaces is called bounded if it maps bounded sets to bounded sets again.

For a locally convex space E are equivalent:

• E is bornological
• Every bounded operator in a further locally convex space F is continuous. ${\displaystyle E\rightarrow F}$

A linear operator is said to be sequential if from in E always follows in F. In non-metrizable spaces this condition can really be weaker than continuity. ${\displaystyle A\colon E\rightarrow F}$ ${\displaystyle \textstyle \lim _{n\in {\mathbb {N} }}x_{n}=x}$ ${\displaystyle \textstyle \lim _{n\in {\mathbb {N} }}Ax_{n}=Ax}$

For a bornological space E and a linear operator are equivalent: ${\displaystyle A:E\rightarrow F}$

• A is continuous.
• A is sequential.
• A is restricted.

Bornological spaces as inductive limits of standardized spaces

A locally convex space E is called an inductive limit of normalized spaces if there are linear mappings with normalized spaces , so that and the topology on E is the finest locally convex topology that makes everything continuous. ${\displaystyle j_{\alpha }\colon E_{\alpha }\rightarrow E}$ ${\displaystyle E_{\alpha }}$ ${\displaystyle \textstyle E=\bigcup _{\alpha }j_{\alpha }(E_{\alpha })}$ ${\displaystyle j_{\alpha }}$

For a locally convex space E are equivalent:

• E is bornological.
• E is an inductive limit of normalized spaces.

One can even specify such an inductive limit. For a bounded and absolutely convex set let . Then there is a vector space, and the Minkowski functional zu makes this vector space a normalized space. The locally convex space is bornological if and only if it bears the inductive locally convex topology of all inclusions , whereby the restricted, absolutely convex sets run through. ${\displaystyle B\subset E}$ ${\displaystyle \textstyle E_{B}:=\bigcup _{\lambda >0}\lambda \cdot B}$ ${\displaystyle E_{B}}$ ${\displaystyle p_{B}}$ ${\displaystyle B\subset E_{B}}$ ${\displaystyle E}$ ${\displaystyle (E_{B},p_{B})\subset E}$ ${\displaystyle B}$

If one can even find a representation for E as an inductive limit of Banach spaces , then E is called ultrabornological . In such spaces the theorem about the open mapping and the theorem about the closed graph apply .

Completeness of the dual space

If E is a locally convex vector space, then every bounded set B in E defines a semi-norm on the dual space by setting. Provided with the set of semi-norms , where B runs through the bounded sets of E, becomes a locally convex vector space, which is then denoted by. This generalizes the dual space formation in normalized spaces . As in the theory of normalized spaces, the following theorem applies: ${\displaystyle P_{B}}$ ${\displaystyle E\,'}$ ${\displaystyle p_{B}(f):=\sup\{|f(x)|;x\in B\}}$ ${\displaystyle p_{B}}$ ${\displaystyle E\,'}$ ${\displaystyle E_{b}'}$

If E is bornological, then it is complete; H. every Cauchy network converges. ${\displaystyle E_{b}'}$

literature

• Klaus Floret, Joseph Wloka : Introduction to the theory of locally convex spaces (= Lecture Notes in Mathematics. Vol. 56, ISSN  0075-8434 ). Springer, Berlin et al. 1968, doi : 10.1007 / BFb0098549 .
• 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 .