George Bachman

Rings of continuous functions with values in a topological field

Let F be a complete topological field. We undertake a study of the ring C(X, F) of all continuous F-valued functions on a topological space X whose topology is determined by C(X, F), in that it is the weakest making each function in C(X, F) continuous, and of the ring C*(X, F) of all continuous F-valued functions with relatively compact range, where the topology of X is similarly determined by C*(X, F). The theory of uniform structures permits a rapid construction of the appropriate generalizations of the Hewitt realcompactification of X in the former case and of the Stone-Cech compactification of X in the latter. Most attention is given to the case where F and X are ultraregular; in this case we determine conditions on F that permit a development parallel to the classical theory where F is the real number field. One example of such conditions is that the cardinality of F be nonmeasurable and that the topology of F be given by an ultrametric or a valuation. Measure-theoretic interpretations are given, and a nonarchimedean analogue of Nachbin and Shirotas theorem concerning the bornologicity of C(X) is obtained. Our goal is to develop a theory parallel to that of realcompact spaces and realcompactifications by replacing the topological field R of real numbers with a complete topological field F, with emphasis on the case where F is ultraregular. (An ultraregular space is a Hausdorff space for which the clopen sets form a basis, a clopen set being one that is both open and closed. An ultranormal space is a Hausdorff space each closed subset of which has a fundamental system of neighborhoods consisting of clopen sets. There exist ultraregular spaces that are not ultranormal [25], [261.) Our emphasis on ultraregular fields is prompted by the fact that every field whose topology is given by a valuation or an ultrametric is ultraregular (and, in fact, is ultranormal). If X and Y are topological spaces, C(X, Y) denotes the set of all continuous functions from X into Y, and C*(X, Y) denotes the set of all continuous functions f from X into Y such that f(X) is compact. If F is a topological field, C(X, F) and C*(X, F) are both F-algebras; if, moreover, F is locally compact, Received by the editors July 24, 1973. AMS (MOS) subject classifications (1970). Primary 54D35, 54D60, 54E15.

Geometry in certain finite groups

Certain finite groups can be considered in a natural manner as geometrical groups in the sence that they, along with a class of conjugate subgroups, characterize a certain geometry and also act as groups of motions on this geometry. Two special classes of such groups are studied, and their structural properties are determined. Finally, examples of such groups are constructed.

Lattice repleteness and some of its applications to topology

We deal with the general concept of lattice repleteness. Specifically, we systematize the study of several important special cases of repleteness, namely, realcompactness, α-completeness, N-compactness, and Borel-completeness; we apply our general results on repleteness to specific lattices in topological spaces, in particular, to analytic spaces; we utilize the concept of Gδ-closure to obtain necessary or sufficient conditions for repleteness (this portion of our work generalizes important theorems of Mrowka on Stone-Cechcompactification, of Frolik on realcompact spaces, and of Wenjen on realcompact spaces); finally, we extend the measure representation material of Varadarajan and then we utilize the results to obtain further applications to repleteness.

Some applications of the adjoint to lattice regular measures

SummaryIn this paper, the principal role is played by the adjoint of a certain bounded linear mapping, whose domain and range are Banach spaces of lattice regular measures. First, the general properties of the adjoint are investigated and it is shown, in particular, how this mapping yields generalizations of many results in Stone-Čech Theory, especially matters related to embeddibility. Then, the investigation continues with the mapping properties of the adjoint, and a variety of applications is given to Topological Measure Theory, strong measure repleteness, tightness, and relative compactness.

The Fourier Transform

Note. In this chapter, unless otherwise indicated, all functions are complex-valued functions of a real variable. Given integrable functions of t, f, and k the function

The Discrete and Fast Fourier Transforms

g\left( \omega \right) = \int {_{E}f\left( t \right)k} \left( {t,\omega } \right)dt

Functional Analysis and Valuation Theory

dt for some set E is called and INTEGRAL TRANSFORM of f with KERNEL k ( t, w) (of the transform). By “transforming” both side of certain equations, we can sometimes convert them into simpler ones—differential equations to algebraic equations, for examples.

On regular extensions of measures.

It is natural to think about distance between physical objects—people, say, or buildings or stars. In what follows, we explore — notion of “closeness” for such things as functions and sequences. (How far is f (x)= x3 from g(x) =sin x? How far is the sequence (1/n) from (2/n2)?) The way we answer such — question is through the idea of — metric space.In principle, it enables us to talk about the distance between colorsor ideas or songs. When we can measure “distance,” we can take limits or “perform analysis.” Special distance-measuring devices called norms are introduced for vector spaces. The analysis we care most about in this book involves norms. This type of analysis is known as functional analysis because the vector spaces of greatest interest are spaces of functions.