## Topological convex spaces |

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abelian action absorbing set affine space affine subset affine transformations bounded affine functional Caratheodory chapter closed convex subset closed set closed subset common fixed point compact convex set compact convex space compact convex subset compact set compact topological convex containing continuous affine mapping continuous transitive action convex hull convex neighborhood convex topological convex corollary of theorem Definition extreme point family of affine finite subset functional f functionals of Rn identity mapping Krein-Milman theorem Let f let g Let us prove Let x e Let xeK linear mapping locally convex topological lower semi lower semi-continuous affine open half space point of A(K Proof proposition relation of equivalence semi-continuous affine functional set of affine set of extreme set of Rn signed measure smallest extremal subset of Rn suffices to prove topological convex space topological vector spaces vague topology VxeC VxeK weak topology yef(x Zorn lemma