Browsing by Author "Golightly, George O."
Now showing 1 - 2 of 2
- Results Per Page
- Sort Options
Item Graph-dense linear transformations in Hilbert space(1977) Golightly, George O.; Nerney, John S. Mac; Murray, Christopher B.; Cook, Howard; Lloyd, Justin T.; Wheeler, Lewis T.In an investigation in 1961 of closed linear relations T in a complete inner product space, Arens introduced the concept of the closed part of T. I have extended this concept to the instance in which T is a linear function from a dense linear subspace of a complete inner product space to a complete inner product space. Moreover, in this setting, T is seen to be naturally decomposed into the sum of its closed part and another, "graph-dense", part, a linear function dense in the Cartesian product of its initial and final sets. My additional investigation is concerned with two principal questions. The first of these relates to the problem of viewing non-closed, unbounded linear functions as arising in a natural fashion. Two processes, one essentially geometric and the other algebraic, for constructing inner products relatively to which a given linear function from a special class is or is not continuous are presented. In the course of the second, for a given linear transformation T and inner product Q, a necessary and sufficient condition that there should exist an inner product Q1 for S such that (S,Q') is continuously included in (S,Q) and T is continuous with respect to the norm for S corresponding to Q' is given. In addition, necessary and sufficient conditions that there should be such a Q' and that (S,Q') be complete are implicitly obtained. These latter conditions involve both (S,Q) and a completion of (S,Q). The second of the two questions involves the extent to which continuity of a function T from a Hilbert space to a Hilbert space is a consequence of linearity. In particular, it is shown, using the decomposition process described initially, that each linear function T from a Hilbert space to an inner product space may be viewed as a linear extension of a densely defined continuous linear function. In the case in which I is graph-dense, a stronger conclusion is drawn.Item Lebesgue-Stieltjes integration(1968) Golightly, George O.; Murray, Christopher B.; MacNerney, John S.; Cook, Howard; Ingram, William T.; Graves, Leon F.In the following, we shall develop enough of the theory of Lebesgue-Stieltjes integration to define the measure of a number set with respect to a non-decreasing function. Our development partially parallels that of F. Riesz's development of the Lebesgue integral.