﻿ LF-space

# LF-space

(LF )-spaces are an viewed in mathematics class of vector spaces. Abstracted to the construction of certain facilities from the distribution theory, one is led informally to the concept of (LF) - space. It is the union of an increasing sequence of Fréchet spaces, which is also called inductive limit of Fréchet spaces, where the name comes from (LF )-space.

## Definition

A (LF )-space is a locally convex space for which there exists a sequence of Fréchet spaces, subject to the following:

This situation is called a representative sequence of Fréchet spaces for. Can you even find a representing sequence of Banach spaces, it is called the space an (LB )-space.

## Examples

Every Fréchet space is an (LF )-space, as a result of performing one can choose the constant sequence.

Be the sequence space of all finite sequences. One identifies with the space of all sequences that have from the - th position only zeros, then a representative result for the (LF) - space, which is even a (LB )-space. The topology is the finest locally convex topology, ie the topology defined by all semi-norms.

The following construction comes from the distribution theory. Is compact, so is the space of all infinitely differentiable functions with support in. Is open so the room is called the space of test functions. Wear it the finest locally convex topology that makes all inclusions steadily. Then, an (LF )-space. As a representative sequence of Fréchet spaces, one can take any sequence, a sequence of compact subsets in is so that each lies in the interior of the union and this is. The topology is independent of the choice of this sequence of compact sets.

## Properties

### Limited quantity

For limited amounts in a (LF )-space with performing sequence following sentence is true:

• A set is bounded if and only if there is a such that and is limited.

### Continuity

The continuity of linear operators from an (LF )-space with performing sequence to another locally convex space can be characterized as follows:

• A linear operator is continuous if all the constraints are continuous.

### Completeness

After going back to Gottfried Köthe set of all (LF )-spaces are complete.

### Relationships to other rooms

(LF )-spaces are barreled ultrabornologisch and have a tissue. Thus, the three classic known from the theory of Banach spaces rates on (LF )-spaces generalize:

Banach - Steinhaus: Is a family of continuous linear operators between locally convex vector spaces, where ( LF) is space, and is limited for each, so is equicontinuous, that is, to every neighborhood of zero, there is a neighborhood of zero, such that for all.

Set over the open picture: A linear, continuous and surjective mapping between (LF) - spaces is open.

Set by the closed graph: A linear mapping between (LF) - spaces with closed graph is continuous.

## Application

In the distribution theory one defines a distribution on an open set as a linear map such that the following continuity condition applies: If compact and is a consequence in so that each carrier has and so uniformly in all leads, then.

This definition is initially not clear whether it ever is continuity with respect to a topology in the continuity condition. It is in fact sufficient to consider sequential continuity, because as (LF )-space bornological. Then the specified condition means nothing else than that all constraints of on, compact, are continuous. After the property to the continuity of linear operators on (LF) - spaces above the continuity with respect to the (LF )-space topology to actually follows.

With the presented concept formation, one can define a distribution as a continuous linear functional on the (LF )-space.

## 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
• Locally convex space
• Functional Analysis
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