Corollary

Tag 07WR, permanent. Cite it as [Bourbaki, Tag 07WR].

Topological Vector Spaces, chapter IV, Appendix 0. Fixed points of groups of affine transformations

— Let E be a reflexive Banach space, G a group of automorphisms of the normed space E, and K a subset of E. Suppose that K is non-empty, convex, closed, bounded and stable under G. Then there exists a point in K which is invariant under G.

Since E is reflexive, K is compact for $\sigma(E, E')$ (IV, p. 15, th. 1). Moreover, every element of G belongs to $\mathscr{L}(E)$ .

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Nothing.

Cites

IV, p. 15