oo f(x) = 00 such that f and g are inverses of each other. Then, interpreting integrals as areas, we see geometrically (draw diagrams in the two cases where b > f(a) and b ::; f(a)) that ab::; F(a) + G(b) , where F(x) = foX f and G(x) = foX 9 for x E lR+ .

The above theorem states that a complete metric space is non-meagre as a subset of itself; a small variation of the proof shows that a locally compact Hausdorff space is also non-meagre as a subset of itself. In applying Baire's theorem, we shall mostly use the following equivalent form of the theorem. 22 (Baire category theorem: second form) Let X be a non-empty, complete metric space, and let (Fn)n2:1 be a sequence of closed subsets such that X = Un2:1 Fn. Then there is some N E N with int FN "I- 0.

34 Introduction to Banach Spaces and Algebras Suppose that F and G are vector subspaces of E. Then F +G = {x + y: x E F, y E G}; we write E = F fJJ G if E = F + G and, further, F n G E / F is a vector space for the operations A(X + F) + /-l(Y + F) = (Ax = {o}. The quotient space + /-lY) + F (x, y E E, A, /-l E K) , and the co dimension of a vector subspace F of E is dim( E / F). Let A and B be subsets of a vector space E, and let A, fL E K. Then A set A in E is convex if sA + tA <;;: A whenever s, t E II with s + t = 1.

### Differentiable Manifolds (1972) by Yozo Matsushima

by Kenneth

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