View source for Kakutani theorem

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055090/k0550901.png" /> be a non-empty compact subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055090/k0550902.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055090/k0550903.png" /> be the set of its subsets and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055090/k0550904.png" /> be an upper [[Semi-continuous mapping|semi-continuous mapping]] such that for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055090/k0550905.png" /> the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055090/k0550906.png" /> is non-empty, closed and convex; then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055090/k0550907.png" /> has a fixed point (i.e. there is a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055090/k0550908.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055090/k0550909.png" />). S. Kakutani showed [[#References|[1]]] that from his theorem the [[Minimax principle|minimax principle]] for finite games does follow.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Kakutani,   "A generalization of Brouwer's fixed point theorem"  ''Duke Math. J.'' , '''8''' :  3  (1941)  pp. 457–459</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  Ky Fan,   "Fixed point and minimax theorems in locally convex topological linear spaces"  ''Proc. Nat. Acad. Sci. USA'' , '''38'''  (1952)  pp. 121–126</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  H. Nikaido,   "Convex structures and economic theory" , Acad. Press  (1968)</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Dugundji,   A. Granas,   "Fixed point theory" , '''1''' , PWN  (1982)</TD></TR></table>