Topologies and projections on Riesz spaces

Hugh Gordon
1960 Transactions of the American Mathematical Society
Introduction. Among the various questions about topologies on Riesz spaces, there is a particularly important one which arises out of Daniell's work on integration. The present paper will be centered around this question, but will not be confined to it. Daniell  considers a space V whose elements are real-valued functions on an abstract set £ and which is closed under the natural linear and lattice operations. He starts with a positive linear functional £ on V (that is, £/^0 whenever/(x) ^0
more » ... ^0 whenever/(x) ^0 for all xEE) endowed with the property that/B(x) 1f(x) pointwise implies £/"-»£/. Daniell extends £ from V to a larger class of functions in such a way that the extended £ satisfies the theorem of Lebesgue: Under suitable conditions of boundedness, if /"-»/ pointwise and the £/B are defined, Ff is defined and £/"->£/. It follows that the extended £ is the integral corresponding to a measure on E. In accordance with the work of Daniell, we call a positive linear functional £ on Fan integral if it has the property: /""["/ pointwise implies Ffn->£/. By an integral we mean any functional which is the difference of two positive integrals. The relationship of the set of integrals to the set of all functionals was considered by Riesz [10, p. 206]. He considered the class of positive functionals which exceed no positive integral other than zero. He showed that any positive functional may be expressed as the sum of a functional of this class and an integral. The result of Riesz employs what is essentially the notion of a direct sum. Of course we may replace that notion by the equivalent one of a projection. The problem is considered from this point of view by Gordon and Lorch 7 J. Suppose the functions in V are bounded. We define a norm on V by |/|| =l.u.b.lSB |/(*)| • Suppose V is a Banach space, i.e. suppose V is complete in its norm. Let V* be the space of all continuous linear functionals on V. Then these authors prove there is a unique projection £ (i.e. an idempotent linear mapping) of V* into itself with the properties that: (a) The range of £ is precisely the set of integrals.