(LF) room

(LF) spaces are a class of vector spaces considered in mathematics . If one abstracts the construction of certain spaces from distribution theory , one is led to the concept of (LF) -space without constraint. It is the union of an ascending sequence of Fréchet spaces , which are also called inductive L imes of F Rechet spaces designated where the name (LF) space stirred.

definition

A (LF) -space is a locally convex space for which there is a sequence of Fréchet spaces such that the following applies: ${\displaystyle E}$ ${\displaystyle (E_{n})_{n}}$

1. ${\displaystyle E_{n}\subset E_{n+1}}$ for all ${\displaystyle n\in {\mathbb {N} }}$
2. For each contributes by given subspace topology . ${\displaystyle n\in {\mathbb {N} }}$ ${\displaystyle E_{n}}$ ${\displaystyle E_{n+1}}$
3. ${\displaystyle E}$is the union of all . ${\displaystyle E_{n}}$
4. ${\displaystyle E}$carries the finest locally convex topology that makes all inclusions continuous. ${\displaystyle E_{n}\subset E}$

In this situation one calls a performing sequence of Fréchet spaces for . If one can even find a descriptive sequence from Banach spaces , the space is called an (LB) space. ${\displaystyle (E_{n})_{n}}$ ${\displaystyle E}$

Some authors also weaken the second condition and only require that the inclusion from to is continuous. For such more general (LF) -spaces not all of the properties given below are automatically fulfilled, in particular there are then (LF) -spaces that are not complete. ${\displaystyle E_{n}}$ ${\displaystyle E_{n+1}}$

Examples

Every Fréchet space is an (LF) space; the constant sequence can be chosen as the representative sequence . ${\displaystyle E}$ ${\displaystyle E_{n}=E}$

Let be the sequence space of all finite sequences. If one identifies with the space of all sequences that have only zeros from the -th position, then there is a descriptive sequence for the (LF) -space , which is even a (LB) -space. The topology on is the finest locally convex topology, i.e. H. the topology defined by all semi-norms . ${\displaystyle c_{00}}$ ${\displaystyle {\mathbb {K} }^{n}}$ ${\displaystyle (n+1)}$ ${\displaystyle ({\mathbb {K} }^{n})_{n}}$ ${\displaystyle c_{00}}$ ${\displaystyle c_{00}}$

The following construction comes from distribution theory . Is compact , so is the space of all infinitely differentiable functions with support in . Is open , so called space the space of test functions on . wear the finest locally convex topology that makes all inclusions continuous. Then there is an (LF) -space. As a descriptive sequence of Fréchet spaces one can take any sequence , where a sequence of compact subsets is in, so that each lies in the interior of and is the union of these . The topology on is independent of the choice of this sequence of compact sets. ${\displaystyle K\subset {\mathbb {R} }^{m}}$ ${\displaystyle C^{\infty }(K)}$ ${\displaystyle K}$ ${\displaystyle \Omega \subset {\mathbb {R} }^{m}}$ ${\displaystyle {\mathcal {D}}(\Omega ):=\bigcup \{C^{\infty }(K);\,K\subset \Omega \,\,{\text{kompakt}}\}}$ ${\displaystyle \Omega }$ ${\displaystyle {\mathcal {D}}(\Omega )}$ ${\displaystyle C^{\infty }(K)\subset {\mathcal {D}}(\Omega )}$ ${\displaystyle {\mathcal {D}}(\Omega )}$ ${\displaystyle (C^{\infty }(K_{n}))_{n}}$ ${\displaystyle (K_{n})_{n}}$ ${\displaystyle \Omega }$ ${\displaystyle K_{n}}$ ${\displaystyle K_{n+1}}$ ${\displaystyle \Omega }$ ${\displaystyle K_{n}}$ ${\displaystyle {\mathcal {D}}(\Omega )}$

properties

Limited quantities

For limited sets in a (LF) -space with a descriptive sequence , the following theorem applies: ${\displaystyle (E_{n})_{n}}$

• A lot is bounded if and when one is so and in is limited. ${\displaystyle B\subset E}$ ${\displaystyle n\in {\mathbb {N} }}$ ${\displaystyle B\subset E_{n}}$ ${\displaystyle B}$ ${\displaystyle E_{n}}$

continuity

The continuity of linear operators from one (LF) -space with a descriptive sequence into another locally convex space can be characterized as follows: ${\displaystyle E}$ ${\displaystyle (E_{n})_{n}}$ ${\displaystyle F}$

• A linear operator is continuous if and only if all restrictions are continuous. ${\displaystyle T:E\rightarrow F}$ ${\displaystyle T|_{E_{n}}:E_{n}\rightarrow F}$

completeness

According to a sentence going back to Gottfried Köthe , all (LF) -spaces are complete .

Relationships with other spaces

(LF) rooms are barreled , ultrabornological and have a fabric . With this, the three classical theorems known from the theory of Banach spaces generalize to (LF) spaces:

Banach-Steinhaus's theorem : Ifa family of continuous linear operatorsbetween locally convex vector spaces, where(LF) -space is, and isboundedfor each, then isequidistant, i.e. H. for every zero neighborhoodthere is a zero neighborhood, so thatfor all. ${\displaystyle (T_{\alpha })_{\alpha \in I}}$ ${\displaystyle E\rightarrow F}$ ${\displaystyle E}$ ${\displaystyle \{T_{\alpha }(x);\alpha \in I\}}$ ${\displaystyle x\in E}$ ${\displaystyle (T_{\alpha })_{\alpha \in I}}$ ${\displaystyle V\subset F}$ ${\displaystyle U\subset E}$ ${\displaystyle T_{\alpha }(U)\subset V}$ ${\displaystyle \alpha \in I}$

Theorem about the open mapping : A linear, continuous and surjective mappingbetween (LF) -spaces is open. ${\displaystyle T:E\rightarrow F}$

Theorem of the closed graph : A linear mappingbetween (LF) -spaces with a closed graph is continuous. ${\displaystyle T:E\rightarrow F}$

application

In distribution theory, a distribution on an open set is defined as a linear mapping , so that the following continuity condition applies: Is compact and is a sequence in , so that every carrier has in and so that is uniform in all derivatives, so is . ${\displaystyle \Omega \subset {\mathbb {R} }^{m}}$ ${\displaystyle T:{\mathcal {D}}(\Omega )\rightarrow {\mathbb {R} }}$ ${\displaystyle K\subset \Omega }$ ${\displaystyle (f_{n})_{n}}$ ${\displaystyle {\mathcal {D}}(\Omega )}$ ${\displaystyle f_{n}}$ ${\displaystyle K}$ ${\displaystyle f_{n}\to 0}$ ${\displaystyle T(f_{n})\to 0}$

With this definition it is initially not clear whether the continuity condition is at all continuity with respect to a topology. Indeed, it is sufficient to consider sequential continuity, because the (LF) -space is bornological . Then the given condition means nothing else than that all restrictions of on , compact, are continuous. After the above-mentioned property of the continuity of linear operators on (LF) -spaces, the continuity with regard to the (LF) -space topology actually follows . ${\displaystyle {\mathcal {D}}(\Omega )}$ ${\displaystyle T}$ ${\displaystyle C^{\infty }(K)}$ ${\displaystyle K\subset \Omega }$ ${\displaystyle {\mathcal {D}}(\Omega )}$

With the concepts presented here, one can define a distribution as a continuous linear functional on the (LF) -space . ${\displaystyle {\mathcal {D}}(\Omega )}$

swell

• K. Floret, J. Wloka: Introduction to the Theory of Locally Convex Spaces , Lecture Notes in Mathematics 56, 1968
• F. Treves: Topological Vector Spaces, Distributions and Kernels , Dover 2006, ISBN 0-486-45352-9