Functional Algebra & Hypercalculus in Infinite Dimensions: Hyperintegrals, Hyperfunctionals & Hyperderivatives (Theoretical and Applied Mathematics)

The theory of hypernumbers and extra functions is further development in distribution theory inspired by contemporary physics and influenced by problems in mathematical physics. It makes more functions differentiable and provides new kinds of derivatives and hyper derivatives aimed at solving more differential and operator equations than ever before possible. In the book, extra functions are extended to hyper functionals and hyperoperators in infinite-dimensional vector spaces. Due to its development, many problems in contemporary physics, as well as in modern linear and nonlinear analysis have an infinite-dimensional nature, and the infinite-dimensional theory of extra functions, hyper functionals and hyperoperators provides new tools for solving many of these problems. The book describes new mathematical structures such as hyper derivatives and hyper integrals of real and complex functions, hyper probability and hyper expectation of random processes and some others, essentially increasing power of functional analysis and probability applications. It presents the key parts of calculus number systems, function spaces, the differential calculus and the integral calculus in the setting of hypernumbers, extra functions, hyper functionals and hyperoperators in finite-dimensional and infinite-dimensional vector spaces. In addition, functional algebra, which employs algebraic operations with extra functions, hyper functionals and hyperoperators is developed. New relations between hyper differentiation and continuity of functions and operators are explicated. As differentiation and integration are special cases of hyper differentiation and hyper integration, respectively, hyper calculus includes calculus as its part or subtheory. It is possible to use this book for enhancing traditional courses of calculus for undergraduates, as well as for teaching separate courses for graduate and undergraduate students at colleges and universities. To achieve these goals, exposition in the book goes from simple topics to more and more advanced topics, while proof of some statements are left as exercises for the students.