Weak Topologies of Normed Linear Spaces

Extreme Points of Vector Functions

“…Finally, let Miß, %) be the space of all essentially bounded measurable functions defined on X. It is well known that Af(/i, 5) constitutes the conjugate space of Liu, %) and consequently the unit sphere of Af(/x, %) is bicompact in the weak * topology [3]. Let (®Ln) denote the direct product of L taken with itself » times and take i®Mn) as the direct product of M n times.…”

The space of point homotopic maps into the circle

“…Suppose G is equivalent to R2q(X) for q = 1 and for X a compact, connected space. Conditions (1) and (2) there exists a positive hyperplane G of G' such that G' = G®Re and such that for each a'GG and each aER, there exists â'GG and äER such that for all maximal functionals £ of S, £(o')+ä = |£(a')+a:|. Let X0 be the functional which defined G. We redefine G0 using the character P0 = 70""1(Xo).…”

Characterizations of certain spaces of continuous functions

“…An equivalent definition (see [2]) is that a directed system Fa converges to F if and only if the system of real numbers Fa(b) converges to F(b) for each bEB. In this topology, S is compact, and for fixed b, F(b) is a continuous function on 2.…”

“…If C is a subset and b an element of B, then d(b, C) =infcgc \\b -c\\. We will have occasion to use MooreSmith limits, and the terminology of [2] will be used for this purpose. A directed set is a class of elements {a} together with a transitive relation > with the property that for every ah a2 there exists a such that a>cti and a>a2.…”