Space with fabric

Rooms with fabric are considered in the mathematical discipline of functional analysis. In interaction with the ultrabornological spaces, they allow generalizations of two central theorems from the theory of Banach spaces , that is, the theorem about the open mapping and the theorem about the closed graph . These rooms were introduced by Marc de Wilde in 1969 for precisely this purpose.

The definition is very technical, but in many applications the special technical conditions can be disregarded, since it can be shown that large classes of topological vector spaces have this property and that therefore the generalizations of the mentioned theorems apply, and these are in the applications essential.

Spaces with mesh can be defined for any topological vector spaces . For reasons of simplicity, only locally convex spaces are considered here. The general theory for topological vector spaces is dealt with in the textbook by H. Jarchow given below.

tissue

A tissue in a locally convex space is a family of subsets , where such that: ${\displaystyle E}$ ${\displaystyle C_{n_{1},\ldots ,n_{k}}\subset E}$ ${\displaystyle k\in {\mathbb {N} },(n_{1},\ldots ,n_{k})\in {\mathbb {N} }^{k}}$

1. Every set is absolutely convex and not empty. ${\displaystyle C_{n_{1},\ldots ,n_{k}}}$
2. ${\displaystyle \bigcup _{n=1}^{\infty }C_{n}=E}$.
3. ${\displaystyle \bigcup _{n=1}^{\infty }C_{n_{1},\ldots ,n_{k},n}=C_{n_{1},\ldots ,n_{k}}}$ for all ${\displaystyle k\in {\mathbb {N} },(n_{1},\ldots ,n_{k})\in {\mathbb {N} }^{k}}$
4. For every sequence of natural numbers there is a sequence of positive real numbers such that for each choice of points the series converges. ${\displaystyle (n_{k})_{k}}$ ${\displaystyle (\lambda _{k})_{k}}$ ${\displaystyle \sum _{n=1}^{\infty }\lambda _{k}x_{k}}$ ${\displaystyle x_{k}\in C_{n_{1},\ldots ,n_{k}}}$

The quantities can be imagined as a web that becomes finer and finer as it grows and spans space, which explains the name fabric . ${\displaystyle C_{n_{1},\ldots ,n_{k}}}$ ${\displaystyle k}$

If there is such a tissue in a locally convex space, the space is said to have a tissue or to be a space with tissue . The German term sounds a bit wooden, the English term webbed space cannot be rendered so easily in German.

Permanent properties

Rooms with fabric have very extensive permanent properties:

• If a space is fabricated and a closed subspace, then and the quotient space are also spaces with fabric. ${\displaystyle E}$ ${\displaystyle F\subset E}$ ${\displaystyle F}$ ${\displaystyle E/F}$
• If there is a sequence of locally convex spaces with tissue, the direct product with the product topology is a space with tissue. ${\displaystyle (E_{n})_{n}}$ ${\displaystyle \prod _{n\in {\mathbb {N} }}E_{n}}$
• If there is a sequence of locally convex spaces with tissue, then the direct sum with the final topology is a space with tissue. ${\displaystyle (E_{n})_{n}}$ ${\displaystyle \bigoplus _{n\in {\mathbb {N} }}E_{n}}$

Examples

• Banach spaces have a fabric. If namely is the unit sphere, the data and (regardless of the sequence !) Form a tissue. ${\displaystyle E}$ ${\displaystyle U}$ ${\displaystyle C_{n_{1},\ldots ,n_{k}}:=\min\{n_{1},\ldots ,n_{k}\}\cdot U}$ ${\displaystyle \lambda _{k}:={\frac {1}{k^{2}}}}$ ${\displaystyle (n_{k})_{k}}$

• Since every Fréchet space is a closed subspace of a countable direct product of Banach spaces, it follows from the above permanent properties that Fréchet spaces have a fabric.

• Furthermore, it follows from the above permanent properties that countable inductive limits of Fréchet spaces have a fabric, because these appear as the quotient of the countable direct sums of Fréchet spaces. In particular, LF spaces have a fabric.

• Sequence Complete (DF) rooms are rooms with tissue.

Graph set and openness

For linear operators between spaces with tissue and ultrabornological spaces one can prove the theorem of the closed graph and the theorem of the open mapping.

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

• 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