Then defining boundedness and continuity relative to associated systems of hyperseminorms, we study relations between relative boundedness and relative continuity for mappings of vector spaces with systems of hyperseminorms and systems of hypernorms.Biography Dimitrie Pompeiu attended both primary and secondary school in Dorohoi in Botosani county in northeastern Romania. We also show that semitopological vector spaces are closely related to systems of hyperseminorms. Sufficient and necessary conditions for a hyperpseudometric (hypermetric) to be induced by a hyperseminorm (hypernorm) are found. To study semitopological vector spaces, hypermetrics and hyperpseudometrics are introduced and it is demonstrated that hyperseminorms, studied in previous works of the author, induce hyperpseudometrics, while hypernorms induce hypermetrics.
Semitopological vector spaces are more general than conventional topological vector spaces, which proved to be very useful for solving many problems in functional. The goal is to provide an efficient base for developing the theory of extrafunction spaces in an abstract setting of algebraic systems and topological spaces. In this paper, we introduce and study semitopological vector spaces. Here our main goal is to find conditions when path hyperintegral has finite value or its value is a real or complex number, i.e., when path hyperintegration coincedes with path integration.
HYPERSPACES MATH SERIES
This better correlates with the situation in contemporary physics, which often encounters infinitely big numbers in a form of divergent series and integrals, while there are no infinitely small number in physics. In contrast to this, the theory of hyperfunctionals and generalized distributions does not change the inner structure of spaces of real and complex numbers, but adds to them infinitely big and oscillating numbers as external objects. For example, nonstandard analysis changes spaces of real and complex numbers by injecting infinitely small numbers and other nonstandard entities. Although, the new theory resembles nonstandard analysis, there are several distinctions between these theories. The theory of hyperfunctionals and generalized distributions, as a part of hyperanalysis that includes hyperintegration, is a novel approach in functional analysis that provides flexible means for analysis in infinite dimensional spaces. It is based on hyperintegration, which extends the path integral to the path hyperintegral. In this paper, a new approach to the path integral is developed.
The Feynman path integral, being very popular in physics, has not yet found a concise unified mathematical representation. Spaces of extrafunctions and hypernumbers are special cases of hyperspaces of integral vector spaces. The main constructions are put together in the context of fiber bundles over hyperspaces of integral vector spaces and integral algebras. In this paper, a method of regularization of irregular operations, functionals and operators is developed and applied to multiplication of hypernumbers and extrafunctions (Section 5) and integration of extrafunctions (Sections 6 and 7). Examples of such operations are multiplication, differentiation and integration, which are important for calculus, differential equations and many applications of mathematics, e.g., in physics. However, there are important operations with functions and operators in function spaces the extension of which by coordinates does not work because their application is not invariant with respect to representations of extrafunctions. Examples of such operations are addition of real functions and multiplication of real functions by real numbers. It is proved that it is possible to extend several basic operations with functions and operators in function spaces to regular operations with extrafunctions and operators in spaces of extrafunctions. Operations and operators performed in this manner are called regular. It is possible to perform some operations with extrafunctions and operators in spaces of extrafunctions applying these operations (operators) separately to each coordinate of the representing sequence.
0 kommentar(er)
