By Alejandro Adem

A few ancient history This publication offers with the cohomology of teams, really finite ones. traditionally, the topic has been certainly one of major interplay among algebra and topology and has at once ended in the production of such very important parts of arithmetic as homo logical algebra and algebraic K-theory. It arose basically within the 1920's and 1930's independently in quantity concept and topology. In topology the main target used to be at the paintings ofH. Hopf, yet B. Eckmann, S. Eilenberg, and S. MacLane (among others) made major contributions. the most thrust of the early paintings right here used to be to attempt to appreciate the meanings of the low dimensional homology teams of an area X. for instance, if the common hide of X was once 3 hooked up, it used to be identified that H2(X; A. ) relies basically at the primary team of X. team cohomology in the beginning looked as if it would clarify this dependence. In quantity thought, team cohomology arose as a traditional equipment for describing the most theorems of sophistication box conception and, specifically, for describing and reading the Brauer crew of a box. It additionally arose obviously within the examine of team extensions, N

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**Example text**

Namely, one can show that it holds provided that V has a dense subset whose cardinality is an Ulam number. Every accessible cardinal is an Ulam number, and the statement that all cardinal numbers are accessible is independent of the usual axioms of set theory. 6] for the definitions of both Ulam numbers and accessible cardinals and for the equivalence between (i) and (ii) under the aforesaid condition. (b) The function f : [0, 1] → L∞ ([0, 1]), given by f (t) = χ[0,t ] , is weakly measurable but not essentially separably valued.

1. g. 7] or [81, Theorem 3, p. 16]) asserts that if μ(X ) < ∞ and if (fi ) is a sequence of real-valued measurable functions on X converging pointwise almost everywhere to a realvalued function f then the sequence converges uniformly to f outside a set of arbitrarily small prescribed measure. 13 shows that, upon passing to a subsequence, the hypothesis that the measure of X is finite can be omitted in the presence of Lp -convergence. 1, we state and prove a vector-valued version of Egoroff’s theorem.

Proof Let (vi∗ ) be a countable dense subset of the dual space V ∗ of a normed space V . Pick a sequence (vi ) ⊂ V such that |vi | ≤ 1 and |vi∗ | ≤ 2 vi∗ , vi for each i . The linear subspace of V spanned by the sequence (vi ) is clearly separable. 2). Assuming that vi∗j → v ∗ in V ∗ , we find that |vi∗j | ≤ 2 vi∗j , vij = 2 vi∗j − v ∗ , vij ≤ 2|vi∗j − v ∗ | → 0. This gives v ∗ = 0, which is a contradiction, and the lemma follows. 4 Every closed subspace of a reflexive Banach space is reflexive.