By Mahlon M. Day (auth.)
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Extra resources for Normed Linear Spaces
Example text
From these examples we can now construct more normed spaces, which shall be called substitution spaces or product spaces. Definition 2. Let 5 be an index set and let X be a Banach space of real-valued functions on 5. If for each s in 5 a normed space N. is given, Jet PxN. be the space of all those functions x on 5 such that (i) x 8 is an element of N. ll for each s in 5, then ~ is in X. (10) If X satisfies the condition that whenever ~EX and ln(s)l:;;;; :;;;; l~(s)l for all s then 17 EX and llnll:;;;; 11~11.
II llx;- x;ll ~ K llx;- x;ll. so (y;) is a norm Cauchy nE4 sequence in B; by completeness (Y;) has a limit y in B. A typical reetangle approximation using the uniformity supplied by K does the rest; for example, if all x.. are in E, then IIT.. x- Yll ~ IITn(X- X;)ll + IIT.. arge; then the middle is made small by taking n far out in ,1. ", B). Theorem 2. Let B and B' be Banach spaces, let L1 be a directed system with a countable cofinal subset, and let (T11 , n E L1) be a net of linear continuous operators from B into B~.
B) Every continuous linear function from an LCS onto a space with the convex core topology is interior. (8) If M is a complete linea,r metric space with distance function d and if a new and always smaller [or always larger] distance function d' is introduced under which M becomes a complete linear metric space M', the identity operator is an isomorphism of M with M'. (9) The theorem of LöwiG that every separable, infinite-dimensional, Banach space has a vector basis of the cardinal nurober of the continuum, and the fact that two such spaces may fail tobe isomorphic (say l 1 ( co) and l 2 (co)), shows that some relation between the metrics is needed in (8) ..