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**Extra info for An Introduction to Sifferentiable Manifolds and Riemannian Geometry**

**Example text**

The form The space of all sequences r for which the sum is in S is denoted by For S a Kbthe space the following statements Theorem. 3 are equivalent: # (a) (b) There is a sequence (tin) is topologically with the relative topology isomorphic to m under the mapping (c) = m and in S such that r(n)un to r; S contains a subspace topologically isomorphic to m in the strong topology; (d) There is a sequence and such that Sa(vn) = with the relative topology 13(S ,S) is topologically isomorphic to 11 under the mapping 1n r(n)vn to r.

The terminology A sequence space S with a linear topology is called a K-space if for each a in A, the domain of the sequences, the mapping S linear functional on S. -- s(a) is a continuous If S is a Frechet space it is called an FK-space; if it is a Banach space it is called a BK-space. A sequence s in a topological sequence space S over N is said to have AK if s[

F). Suppose there is s in S and a continuous semi- p(v) for each assume that p(uv) We may s(n)en) is not 0. norm p on S such that limk in S and 0 There is an increasing sequence of indices Iu(i)l and c > 1. 0 such k. r and 0' s let tu For each sequence u of ones c for each i. n where convergence is oointwise. Then ftul is an uncountable set with p(tu_tv) > for This means S is not separable with respect p. u v. 1. (d) (a). (a) (g) since weakly compact and normal compact subsets coincide in . /1/ The support of a sequence t means the set of indices j which t(j) 0.