Linear Topological Spaces by John L. Kelley

By John L. Kelley

Extra members Are W. F. Donoghue, Jr., Kenneth R. Lucas. B. J. Pettis, Ebbe Thue Poulsen, G. Baley rate, Wendy Robertson, W. R. Scott, And Kennan T. Smith.

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A topological space is regular if and only if for each point x, the family of closed neighborhoods of x is a base for the neighborhood system of x. Thus, X is regular if and only if for each x in X and each neighborhood U of x there is a closed neighborhood V of x such that V c U. A topological space X is separable if and only if there exists a countable subset A of X which is dense in X. If X and Y are topological spaces, then Y is a topological subspace of X if and only if Y c X and the sets which are open in Y are precisely the intersections of Y with open subsets of X.

Iv) This follows from the fact that C + D is the image of the compact set C x D under the continuous map, addition. (v) A point x belongs to A- if and only if for each neighborhood V of 0, x + V intersects A; that is, if and only if x E A - V. Hence A- = A - V: V E 0//}, but V is a neighborhood of 0 if and only if - V is a neighborhood of 0. The required result follows. (vi) Suppose that for each neighborhood V in the neighborhood system Oll of 0 there is an element xv in C suchthat Xv + V intersects the complement of U.

SEC. fE 11 k a positive integer}. Then g is a non-negatively homogeneous, non-negative, subadditive functional on E and g ~ p. E. L'. Chapter 2 LINEAR TOPOLOGICAL SPACES This chapter is largely preliminary in nature; it consists of a brief review of some of the terminology and the elementary theorems of general topology, an examination of the new concept "linear topological space" in terms of more familiar notions, and a comparison of this new concept with the mathematical objects of which it is an abstraction.

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