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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.