By Mahlon Marsh Day
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Additional resources for Normed Linear Spaces, 3rd Edition (Ergebnisse der Mathematik und ihrer Grenzgebiete)
Example text
If ~ and ~' are families of LTSs, say that ~ cgn ~' cni ~'] whenever for every L in ~ and 1; in ~' L cgn 1; [L cni 1;J. (3) If L, L 1 , and 1; are LTSs, if To is a continuous, interior, linear function from L onto L 1 , if Tl is a linear function from Ll into 1;, and if T= Tl To, then T and Tl have precisely the same ones of the following properties: (a) closed, (b) continuous, (c) interior, (d) onto 1;. (4) Let T be linear from one LTS L into another 1;, then (a) if T is closed, T- 1 (O) is also closed.
The closure of a convex set K in an LCS L is determined by the space L*; no more i1iformation about the locally convex topology in L is needed. The proof of the separation theorem for convex sets shows that both Eidelheit and Mazur theorems follow easily from the following simple special case of Cor. 1. Theorem 6 (Existence theorem). If W is a wedge with non-empty interior and W =1= L, then there is a non-trivial monotone f in L*; that is, there is a closed hyperplane H through zero such that W lies on one side of H.
Every linear subspace of a normed space, when provided with the norm induced by the large space, is also a normed space; a norm can be introduced in a factor space of a normed space, as follows: Definition 1. If E is a flat, closed subset of a normed linear space N, let IIEII=inf{lIxll:xEE}. Lemnia 1. If No is a closed linear subspace of N, if Nt =N/No is normed by the definition above, and if To is the natural mapping Tox=x+N o , then (a) Nt is a normed linear space, (b) To is continuous and interior (Def.