By Kenji Ueno, Koji Shiga, Shigeyuki Morita, Toshikazu Sunada

This three-volume set addresses the interaction among topology, services, geometry, and algebra. Bringing the wonder and enjoyable of arithmetic to the school room, the authors provide severe arithmetic in a full of life, reader-friendly type. integrated are workouts and lots of figures illustrating the most techniques. it's compatible for complex high-school scholars, graduate scholars, and researchers.

**Read or Download A Mathematical Gift I, II, III: The Interplay Between Topology, Functions, Geometry, and Algebra (Mathematical World) (v. 1-3) PDF**

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If {Xl' ... } is any basis of L, L is the algebraic direct sum of the subspaces M and N with 22 TOPOLOGICAL VECTOR SPACES [Ch. I bases {Xl' ... , x n- 1 } and {xn}, respectively. By assumption, M is isomorphic with K~-l; since Ko is complete, M is complete and since L is Hausdorff, M is closed in L. 1). 4) that L = M ® N, which means that (Al' ... , An) -+ AIXI + ... + AnXn is an isomorphism of K~-l x Ko = K8 onto L. Finally, it is obvious that every isomorphism of K8 onto L is of this form. 2) may fail for n > 1 when K is not complete (Exercise 4).

6), has a unique continuous extension jj to L that generates the topology of L; (L, jj) is a Banach space. It is obvious that a subspace of a normable space is normable and that a closed subspace of a Banach space is a Banach space. 2 The product of a family of normable spaces is normable if and only if the number offactors :#= {OJ is finite. Proof. 5), a O-neighborhood in the product IT"L", can be bounded if and only if the number of factors L,,:#= {OJ is finite. REMARK. A norm generating the topology of the product of a finite family of normed spaces can be constructed from the given norms in a variety of ways (Exercise 4).

M) in V for which V c m U (YI + P V). 1= 1 We denote by M the smallest subspace of L containing all YI (l = 1, ... , m) and show that M = L, which will complete the proof. 2), is closed in L while {w + An V: n EN} is a neighborhood base of w. Let J1. be any number in K such that w + J1. l. Clearly, b ~ IAnol > O. 0 I ~ 3b/2. opV. oP I ~ 3b/4; hence the assumption M "# L is absurd. 24 TOPOLOGICAL VECTOR SPACES [Ch. I 4. LINEAR MANIFOLDS AND HYPERPLANES If L is a vector space, a linear manifold (or affine subspace) in L is a subset which is a translate of a subspace MeL, that is, a set F of the form Xo + M for some Xo e L.