Lemma of Kakutani

The lemma Kakutani is a mathematical theorem that both the area of the convex geometry as well as that of the functional analysis can be attributed. It goes back to a work by the Japanese mathematician Shizuo Kakutani from 1937 and deals with a property of convex sets in real vector spaces .

Formulation of the lemma

The lemma can be formulated as follows:

Given a real vector space and in two disjoint convex subsets and one situated outside these two quantities point . ${\displaystyle X}$ ${\displaystyle C_{1},C_{2}\subset X}$ ${\displaystyle x\in {X\setminus (C_{1}\cup C_{2})}}$

${\displaystyle {\Gamma }_{i}\subset X\;(i=1,2)}$be the convex hull of . ${\displaystyle \{x\}\cup C_{i}}$

Then:

At least one of the two intersections is the empty set . ${\displaystyle {\Gamma }_{1}\cap C_{2}\;,\;{\Gamma }_{2}\cap C_{1}}$

Corollary: A sentence by Marshall Harvey Stone

From Kakutani 's lemma , with the help of Zorn's lemma, a theorem by Marshall Harvey Stone can be deduced, which Frederick A. Valentine describes as fundamental in his textbook Convex Sets . This sentence can be formulated as follows:

In any real vector space disjoint for every two existing non-empty convex subsets always a separation with comprehensive convex subsets . ${\displaystyle X}$ ${\displaystyle C_{1},C_{2}\subset X}$ ${\displaystyle Z_{1}{\dot {\cup }}Z_{2}=X}$ ${\displaystyle Z_{i}\supset C_{i}\;(i=1,2)}$

As regards the naming is to be noted that Kelley / Namioka the set mentioned as a set of Stone ( English Stone's theorem call) while be seen from the illustration of Valentine rather that the set is to assign Kakutani equally, and probably also by other mathematicians was shown. What is remarkable about Valentine's presentation is the fact that he implicitly uses the Kakutani lemma in the proof, but does not explicitly mention it as such.

Relation to the principle of separation from Eidelheit

Of Gottfried Kothe the set of Stone is called separation theorem called, because it is directly related to the separation rate of Eidelheit ( English : Eidelheit's separation theorem ), which in turn leads to the geometric shape of the set of tap-Banach . The Eidelheitsche separation theorem gave Shizuo Kakutani the occasion for his work from 1937.

Eidelheit's separation theorem can be given convex geometrically as follows:

Let it be a real topological vector space and let it contain two non-empty convex subsets . ${\displaystyle X}$ ${\displaystyle C_{1},C_{2}\subset X}$

${\displaystyle C_{1}}$have inner points , none of which is also a point of . ${\displaystyle C_{2}}$

Then:

(1) There is a closed real hyperplane within and separating it such that none of the inner points of is also a point of . ${\displaystyle X}$ ${\displaystyle C_{1}}$ ${\displaystyle C_{2}}$ ${\displaystyle H\subset X}$ ${\displaystyle C_{1}}$ ${\displaystyle H}$

(2) If both and are open subsets of , they are in different open half-spaces and are strictly separated from each other in this sense . ${\displaystyle C_{1}}$ ${\displaystyle C_{2}}$ ${\displaystyle X}$ ${\displaystyle H}$

In Valentine there is an even more general version of the separation sentence.

literature

• Marcel Berger : Geometry I (=  university text ). Springer Verlag, Berlin / Heidelberg / New York 1987, ISBN 3-540-11658-3 ( MR2724360 ).
• Nicolas Bourbaki : Topological Vector Spaces (=  Elements of Mathematics ). Springer Verlag, Berlin / Heidelberg / New York / London, Paris / Tokyo 1987, ISBN 3-540-13627-4 , Chapters 1–5, pp. II.36 ff . ( MR0910295 ).
• Shizuo Kakutani: A proof of M. Eidelheit's theorem about convex sets . In: Proceedings of the Imperial Academy . tape 13 , 1937, pp. 93-94 ( projecteuclid.org ). MR1568455
• John L. Kelley , Isaac Namioka et al .: Linear Topological Spaces (=  Graduate Texts in Mathematics . Volume 36 ). 2nd Edition. Springer Verlag, New York / Heidelberg / Berlin 1976 ( MR0394084 ).
• Marshall Harvey Stone: Convexity . Reproduced lecture drafting by Harley Flanders . University of Chicago, Chicago 1946.
• Gottfried Köthe: Topological linear spaces I (=  The basic teachings of the mathematical sciences in individual representations with special consideration of the areas of application . Volume 107 ). 2nd improved edition. Springer Verlag, Berlin / Heidelberg / New York 1966, p. 36 ff . ( MR0194863 ).
• Robert E. Megginson: An Introduction to Banach Space Theory (  Graduate Texts in Mathematics . Volume 183 ). Springer Verlag, New York 1998, ISBN 0-387-98431-3 ( MR1650235 ).
• Marshall Harvey Stone: Convexity . Reproduced lecture drafting by Harley Flanders . University of Chicago, Chicago 1946.
• Frederick A. Valentine: Convex sets (=  BI university pocket books. 402 / 402a). Springer Verlag, Mannheim 1968 ( MR0226495 ).

Individual evidence

1. ↑ ^{a b} Marcel Berger: Geometry I. 1987, p. 384
2. ^ ^{A b c d} John L. Kelley, Isaac Namioka: Linear Topological Spaces. 1976, p. 17
3. ^ ^{A b} Frederick A. Valentine: Convex sets. 1968, pp. 29-30
4. ↑ Valentine, op.cit., P. 29
5. ↑ Valentine, op.cit., P. 30
6. ↑ Gottfried Köthe: Topological linear spaces I. 1966, p. 189 ff
7. ^ Nicolas Bourbaki: Topological Vector Spaces. 1998, II.36 ff
8. ^ ^{A b} Robert E. Megginson: An Introduction to Banach Space Theory. 1998, p. 179
9. ↑ Shizuo Kakutani: A proof of M. Eidelheit's theorem about convex sets. in: Proc. Imp. Acad. 13, p. 93
10. ↑ Köthe, op.cit., P. 191
11. ↑ Bourbaki, op. Cit., II.37
12. ↑ Valentine, op.cit., P. 34