By G.L.M. Groenewegen, A.C.M. van Rooij

The house C(X) of all non-stop services on a compact house X consists of the constitution of a normed vector house, an algebra and a lattice. at the one hand we learn the family members among those buildings and the topology of X, however we speak about a few classical effects in accordance with which an algebra or a vector lattice should be represented as a C(X). quite a few functions of those theorems are given.Some recognition is dedicated to comparable theorems, e.g. the Stone Theorem for Boolean algebras and the Riesz illustration Theorem.The publication is practical analytic in personality. It doesn't presuppose a lot wisdom of useful research; it includes introductions into matters similar to the vulnerable topology, vector lattices and (some) integration theory.

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**Example text**

Thus, let ???? ∈ [−1,1]. Defining ????: ℝ ⟶ ℝ by 1 ????(????) ≔ ???? − 2(???? 2 − ????2 ) (???? ∈ ℝ), we have ????????+1 (????) = ????(???????? (????)). Note that ????(|????|) = |????|, ????(0) ≥ 0 and ???? is increasing on the interval [0, |????|]. It follows that ???? is an increasing map [0, |????|] ⟶ [0, |????|]. Then the numbers 0, ????(0), ????(????(0)), … form an increasing sequence in [0, |????|], converging to a point ???? of [0, |????|] with ????(????) = ????. This means precisely that (???????? (????))???? is an increasing sequence in [0, |????|], converging to a point ???? of [0, |????|] that satisfies ????2 = ????2 and therefore must be |????|.

This fact makes complete regularity fundamental for our theory. Throughout this chapter, ???? is a set and ???? is a collection of functions on ????. Weak Topologies. Complete Regularity. 1 There are two “weak topologies” associated with the situation sketched above, one on ???? and one on ????. Both are easy to describe in terms of convergence of nets. ) In ???? , a net (???????? )????∈???? “converges weakly” to ???? if and only if ???????? (????) ⟶ ????(????) for every ???? in ????. In ???? , a net (???????? )????∈???? “converges weakly” to ???? if and only if ???????? (????) ⟶ ????(????) for every ???? in ????.

44 5 Riesz Spaces For ???? in ????(????) the function |????|, as above, is continuous and hence is the smallest upper bound of {????, −????} in the lattice ????(????). Thus, ????(????) is a Riesz space, and for ???? in ????(????), |????| is the absolute value of ???? relative to the Riesz space structure of ????(????). (4) The above may look excessively pedant, but consider the following. Call a function ????: [0,1] ⟶ ℝ “affine” if there exist numbers ???? and ???? such that ????(????) = ???? + ???????? (???? ∈ [0,1]).