Uniform Structures on Topological Groups and Their Quotients by W. Roelcke

By W. Roelcke

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Then there exists an element a ∈ A \ B such that B ∪ {a} is still independent (augmentation or exchange property). A maximal independent set is called a basis. It follows immediately from axiom (c) that all bases have the same cardinality, called the rank of the matroid. If M is a matroid on the ground set E, we can define its dual matroid M ∗ , which has the same ground set and whose bases are given by the set complements E \ B of bases B of M . We have rank(M ∗ ) = |E| − rank(M ). We shall mainly be concerned with the matroid structure of a list of vectors.

We claim that W ∩ A is a face. In fact, it is convex, and if [a, b] is a segment in A meeting W ∩ A in a point q ∈ (a, b), we claim that [a, b] ⊂ W . Otherwise, a, b ∈ / W, and let W be the affine space spanned by W and a. Then W is a hyperplane in W and a, b lie in the two distinct half-spaces in which W divides W . By maximality there is a point ˚ and we may assume that it is in the same half-space as a. 6 (i) the segment (r, b) is contained in A, W, giving a contradiction. 20. If a closed convex set A does not contain any line then A is the convex envelope of its extremal points and extremal rays.

1) C(X) is a convex polyhedral cone. ˆ (2) C(X) is a pointed cone if and only if X spans V . ˆ ˆ (3) C(X) = C(X). ˆ Proof. The fact that C(X) is a polyhedral cone is clear by definition. ˆ The cone C(X) contains a line if and only if there is a nonzero φ ∈ V ∗ ˆ with both φ and −φ lying in C(X). But this holds if and only if φ vanishes on C(X), that is, C(X) lies in the hyperplane of equation φ| v = 0. ˆ ˆ Finally, C(X) ⊃ C(X). 8 there exist a φ ∈ V ∗ and a constant a such that φ| p a, while φ| v ≤ a for v ∈ C(X).

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