By Ronald Larsen
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And the coincide. Therefore part (ii) of the theorem implies part (i). 0 This is not the most general theorem about the metrization of topological linear spaces. linear space Indeed, one can show that a topological is metrizable if and only if there exists a (V,T) countable neighborhood base at the origin for T. The metric can also be constructed in this case so as to be translation invariant. For details the reader is referred to [K, pp. 162-164; KeNa, pp. 48-49]. 2_6. Problems. 1. over L. 2) Let (V,T) be a topological linear space Prove that for each y E V and each yy : V V, defined by a E f, a # 0, yy(x) = x + y, x E V, and the 2.
O The reader should note the use made here of the fact that not all the seminorms in P vanish at any one nonzero vector in Given a seminormed linear space topology generated by UP as Ti,, (V,P) V. 5 25 Topology in Seminormed Linear Spaces It is evident that this topology by the base for at TP When UP(x) is a x. is a normed linear space, the neighborhoods in (V,P) UP(x) are, of course, just open balls with center ology TP coincides with the norm topology. x, and the top- In the general case of a seminormed linear space it is not sufficient, however, to use only neighborhoods of of of the form x U(x,c,p), p E P, as the elements since the intersection of two such sets need not contain Up(x) Hence there are not enough such sets to form a a third such set.
Is =lxk11 = 0, The series Prove that a is complete if and only if every abso- lutely summable series is summable. 0. Introduction. space (V,P) We have seen that, given a seminormed linear we can introduce a Hausdorff topology Tp into that is intimately connected with the family of seminorms V Now we P. wish to study such spaces -- that is, linear spaces equipped with a Hausdorff topology -- in their own right. In order to make these topological linear spaces interesting objects of investigation we shall demand that the linear space operations and the topology "fit together" properly, namely, that the operations be continuous.