Mazur's theorem (convexity and compactness)

The set of Mazur to convexity and compactness is one of several tenets which the Polish mathematician Stanislaw Mazur the mathematical sub-region of the functional analysis has contributed. The theorem goes back to a work by Mazur from 1930 and deals with a fundamental question of compactness in connection with convex subsets of Banach spaces . From this Mazur's theorem, Schauder's fixed point theorem - in the version for Banach spaces - can be obtained as a consequence.

Formulation of the sentence

The sentence says the following:

Let a Banach space be given and further a subset located in it as well as its closed convex hull . ${\displaystyle X}$ ${\displaystyle T\subseteq X}$ ${\displaystyle K={\overline {\operatorname {conv} {T}}}\supseteq T}$

Then:

Is a compact subset of , then is also such a. ${\displaystyle T}$ ${\displaystyle X}$ ${\displaystyle K}$

generalization

In the textbook by Jürg T. Marti and also in that of AP Robertson and WJ Robertson , Mazur's theorem is formulated even more generally. This can be summarized as follows:

Given a hausdorff shear locally convex topological - vector space as well as a subset . ${\displaystyle \mathbb {R} }$ ${\displaystyle X}$ ${\displaystyle T\subseteq X}$

Then:

If a precompact subset of , then its convex hull , its absolutely convex hull and its closed absolutely convex hull are also precompact subsets. ${\displaystyle T}$ ${\displaystyle X}$ ${\displaystyle \operatorname {conv} {T}}$ ${\displaystyle \Gamma {T}}$ ${\displaystyle {\overline {\Gamma {T}}}}$

Further tightening in Euclidean space

In Euclidean space it even applies:

For any compact subset , the convex hull is (itself) compact. ${\displaystyle T\subset \mathbb {R} ^{n}\;(n\in \mathbb {N} )}$ ${\displaystyle \operatorname {conv} {T}}$

Notes and explanations

• The precompactness of a subset is to be understood here in relation to the uniform structure induced by the zero neighborhood basis of . Such a subset is therefore precompact if and only if there are finitely many points for every zero neighborhood , so that the coverage is given. ${\displaystyle T\subset X}$ ${\displaystyle X}$ ${\displaystyle W\subseteq X}$ ${\displaystyle x_{1},x_{2},\ldots ,x_{m}\in X\;(m\in \mathbb {N} )}$ ${\displaystyle \bigcup _{j=1}^{m}{\left(x_{j}+W\right)}\supseteq T}$
• In every metric space - including every Banach space - a subset is precompact if and only if its closed envelope is precompact. Here a subset is relatively compact if it is precompact and its closed shell is complete . In a Banach space, a subset is precompact if and only if it is relatively compact.
• Is restricted, then applies . ${\displaystyle T\subset \mathbb {R} ^{n}}$ ${\displaystyle \operatorname {conv} {\overline {T}}={\overline {\operatorname {conv} {T}}}}$

literature

• Lothar Collatz : Functional Analysis and Numerical Mathematics . Unchanged reprint of the 1st edition from 1964 (=  The Basic Teachings of Mathematical Sciences in Individual Representations . Volume 120 ). 2nd Edition. Springer-Verlag , Berlin, Heidelberg, New York 1968, ISBN 3-540-04135-4 ( MR0165651 ).
• Egbert Harzheim : Introduction to combinatorial topology (=  mathematics. Introductions to the subject matter and results of its sub-areas and related sciences ). Scientific Book Society, Darmstadt 1978, ISBN 3-534-07016-X ( MR0533264 ).
• Friedrich Hirzebruch , Winfried Scharlau : Introduction to functional analysis (=  BI university pocket books . Volume 296 ). Bibliographisches Institut , Mannheim / Vienna / Zurich 1971, ISBN 3-411-00296-4 ( MR1183466 ).
• Kurt Leichtweiß : Convex quantities (=  university text ). Springer-Verlag, Berlin, Heidelberg, New York 1980, ISBN 3-540-09071-1 ( MR0586235 ).
• Jürg T. Marti: Convex Analysis (=  textbooks and monographs from the field of exact sciences, mathematical series . Volume 54 ). Birkhäuser Verlag , Basel, Stuttgart 1977, ISBN 3-7643-0839-7 ( MR0511737 ).
• S. Mazur: About the smallest convex set that contains a given compact set . In: Studia Mathematica . tape 2 , 1930, p. 7-9 .
• Albrecht Pietsch : History of Banach Spaces and Linear Operators . Birkhäuse, Boston, Basel, Berlin 2007, ISBN 0-8176-4367-2 ( MR2300779 ).
• AP Robertson, WJ Robertson: Topological vector spaces . Translation from English by Horst S. Holdgrün (= BI university paperbacks  . 164 / 164a). Bibliographisches Institut, Mannheim 1967 ( MR0209926 ).

Individual evidence

1. ^ Lothar Collatz: Functional Analysis and Numerical Mathematics. 1968, p. 352 ff, p. 359
2. ^ ^{A b} Albrecht Pietsch: History of Banach Spaces and Linear Operators. 2007, p. 74
3. ↑ Collatz, op.cit., P. 355
4. ↑ Collatz, op.cit., P. 352
5. ↑ Jürg T. Marti: Convex Analysis. 1977, p. 23
6. ^ AP Robertson, WJ Robertson: Topological vector spaces. 1967, p. 61
7. ↑ However, the name of Stanisław Mazur is not mentioned in Robertson / Robertson, while Marti explicitly refers to Mazur.
8. ↑ Egbert Harzheim: Introduction to combinatorial topology. 1978, p. 25
9. ↑ Kurt Leichtweiß: Convex sets. 1980, p. 24
10. ↑ ^{a b} Marti, op.cit., P. 202
11. ^ Robertson / Robertson, op.cit., P. 58
12. ↑ Friedrich Hirzebruch, Winfried Scharlau: Introduction to Functional Analysis. 1971, p. 18