Weighted Space of Measurable Functions

Key definitions and results in symmetric spaces, particularly Lp, Lorentz, Marcinkiewicz and Orlicz spaces are emphasized in this textbook. A comprehensive overview of the Lorentz, Marcinkiewicz and Orlicz spaces is presented based on concepts and results of symmetric spaces. Scientists and researchers will find the application of linear operators, ergodic theory, harmonic analysis and mathematical physics noteworthy and useful.

The principal aim of this book is to introduce to the widest possible audience an original view of belief calculus and uncertainty theory. In this geometric approach to uncertainty, uncertainty measures can be seen as points of a suitably complex geometric space, and manipulated in that space, for example, combined or conditioned.

This text is appropriate for a one-semester course in what is usually called ad­ vanced calculus of several variables. The approach taken here extends elementary results about derivatives and integrals of single-variable functions to functions in several-variable Euclidean space. The elementary material in the single- and several-variable case leads naturally to significant advanced theorems about func­ tions of multiple variables. In the first three chapters, differentiability and derivatives are defined; prop­ erties of derivatives reducible to the scalar, real-valued case are discussed; and two results from the vector case, important to the theoretical development of curves and surfaces, are presented. The next three chapters proceed analogously through the development of integration theory. Integrals and integrability are de­ fined; properties of integrals of scalar functions are discussed; and results about scalar integrals of vector functions are presented. The development of these lat­ ter theorems, the vector-field theorems, brings together a number of results from other chapters and emphasizes the physical applications of the theory.

This text is appropriate for a one-semester course in what is usually called ad­ vanced calculus of several variables. The approach taken here extends elementary results about derivatives and integrals of single-variable functions to functions in several-variable Euclidean space. The elementary material in the single- and several-variable case leads naturally to significant advanced theorems about func­ tions of multiple variables. In the first three chapters, differentiability and derivatives are defined; prop­ erties of derivatives reducible to the scalar, real-valued case are discussed; and two results from the vector case, important to the theoretical development of curves and surfaces, are presented. The next three chapters proceed analogously through the development of integration theory. Integrals and integrability are de­ fined; properties of integrals of scalar functions are discussed; and results about scalar integrals of vector functions are presented. The development of these lat­ ter theorems, the vector-field theorems, brings together a number of results from other chapters and emphasizes the physical applications of the theory.

In the Part at hand the authors undertake to give a presentation of the historical development of the theory of imbedding of function spaces, of the internal as well as the externals motives which have stimulated it, and of the current state of art in the field, in particular, what regards the methods employed today.