Crane-Milman Theorem

The Kerin – Milman theorem is an important fact from convex analysis in linear topological spaces . It is proved by Mark Grigorievich Krein and David Pinhusovich Milman (1940). ^[one]

Content

1 Formulation
- 1.1 Notes
2 Applications
3 Literature
4 notes

Wording

Let be $L$ ${\ displaystyle L}$ - locally convex space , $K$ ${\ displaystyle K}$ - convex compact in $L$ ${\ displaystyle L}$ , $E$ ${\ displaystyle E}$ - many extreme points $K$ ${\ displaystyle K}$ . Then $K$ ${\ displaystyle K}$ coincides with the closure of the convex hull of the set $E$ ${\ displaystyle E}$ .

Remarks

For infinite-dimensional spaces, this theorem, like many other results, cannot be proved without using the axiom of choice or equivalent statements of set theory.

There are topological vector spaces containing convex compacta without boundary points. ^[2]

Applications

Proof of the nonisomorphism of various Banach spaces .
It is used in an elegant proof of the Stone – Weierstrass theorem belonging to de Branges.

Literature

Kirillov A.A., Gvishiani A.D. Theorems and problems of functional analysis, 1988.

Notes

↑ M. Krein, D. Milman, On extreme points of regular convex sets, Studia Mathematica 9 (1940), 133–138.
↑ Roberts, James W. "A compact convex set with no extreme points." Studia Mathematica 60.3 (1977): 255-266.

[1] M. Krein, D. Milman, On extreme points of regular convex sets, Studia Mathematica 9 (1940), 133–138.

[2] Roberts, James W. "A compact convex set with no extreme points." Studia Mathematica 60.3 (1977): 255-266.