Lasry's theorem

The set of Lasry , also known under the keyword Lasry'sche equation , English Lasry's equality or equality Lasry , is a tenet of the mathematical area of functional analysis and was around the year 1973 by the French mathematician Jean-Michel Lasry presented. The theorem gives an equation, directly related to the inequality of Ky Fan, for certain real functions on topological product spaces .

Formulation of the sentence

Following Jürgen Heine's monograph , the sentence can be formulated as follows:

If one were not empty compact pseudometric room as well as a part-time normalized - vector space and is a non-empty convex subset with associated than the amount of continuous maps . ${\displaystyle X}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle V}$ ${\displaystyle Y\subseteq V}$ ${\displaystyle C(X,Y)}$

Let a real function be given . ${\displaystyle f\colon X\times Y\to \mathbb {R} }$

It should on the one hand for each of the subordinate function concave and bounded from above and on the other hand for each of the subordinate function lower continuous . ${\displaystyle x\in X}$ ${\displaystyle f(x,{\cdot })\colon Y\to \mathbb {R} \;,\;y\mapsto f(x,y)\;,\;}$ ${\displaystyle y\in Y}$ ${\displaystyle f({\cdot },y)\colon X\to \mathbb {R} \;,\;x\mapsto f(x,y)\;,\;}$

Then applies

${\displaystyle \inf _{x\in X}\sup _{y\in Y}{f(x,y)}=\sup _{\phi \in C(X,Y)}\inf _{x\in X}{f(x,{\phi (x)})}}$.

General background

In connection with Lasry's Theorem, the following general result is significant:

Given are non-empty sets and with as the associated set of mappings and a real function . ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle Y^{X}}$ ${\displaystyle f\colon X\times Y\to \mathbb {R} }$

The subordinate function is limited upwards for each . ${\displaystyle x\in X}$ ${\displaystyle f(x,{\cdot })\colon Y\to \mathbb {R} }$

Then applies

${\displaystyle \inf _{x\in X}\sup _{y\in Y}{f(x,y)}=\sup _{\phi \in Y^{X}}\inf _{x\in X}{f(x,{\phi (x)})}}$.

annotation

Lasry's theorem says that in the situation mentioned there the corresponding equation with is correct instead of . ${\displaystyle C(X,Y)}$ ${\displaystyle Y^{X}}$

literature

• Jürgen Heine: Topology and Functional Analysis . Basics of abstract analysis with applications. 2nd, improved edition. Oldenbourg Verlag , Munich 2011, ISBN 978-3-486-70530-0 .
• Jean-Pierre Aubin : Optima and Equilibria . An introduction to nonlinear analysis. Translated from the French by Stephen Wilson (=  Graduate Texts in Mathematics . Volume 140 ). 2nd Edition. Springer Verlag, Berlin 1998, ISBN 3-540-64983-2 ( MR1729758 ).
• Jean-Pierre Aubin: Applied Abstract Analysis . Exercises by Bernard Cornet and Hervé Moulin. Translated from the French by Carole Labrousse (=  Pure and Applied Mathematics ). John Wiley & Sons , New York, London, Sydney, Toronto 1977, ISBN 0-471-02146-6 ( MR0470034 ).

See also

• Min-Max Theorem

Individual evidence

1. ↑ Jean-Michel Lasry teaches (among other things) at the University of Paris-Dauphine . For more information, see the personal article about Lasry on French Wikipedia !
2. ↑ Jürgen Heine: Topology and Functional Analysis. 2011, p. 296 ff.
3. ^ Jean-Pierre Aubin: Applied Abstract Analysis. 1998, p. 199 ff.
4. ^ Jean-Pierre Aubin: Optima and Equilibria. 1998, p. 137 ff.
5. ↑ Prof. Dr. Jürgen Heine taught at the Institute for Applied Mathematics at the University of Hanover .
6. ↑ ^{a b} Heine, op.cit., P. 297.
7. ↑ Heine, op.cit., P. 296.