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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.
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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.
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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.
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Operations and operators performed in this manner are called regular. Among other things, the following results are presented: (1) If ( X, d ) is a metric space, then the following conditions are equivalent: (a) ( X, f ) is weakly mixing, (b) ( ( F ( X ), d ∞ ), f ^ ) is transitive, (c) ( ( F ( X ), d 0 ), f ^ ) is transitive and (d) ( ( F ( X ), d S ) ), f ^ ) is transitive, (2) if f : ( X, d ) → ( X, d ) is a continuous function, then the following hold: (a) if ( ( F ( X ), d S ), f ^ ) is transitive, then ( ( F ( X ), d E ), f ^ ) is transitive, (b) if ( ( F ( X ), d S ), f ^ ) is transitive, then ( X, f ) is transitive and (3) if ( X, d ) be a complete metric space, then the following conditions are equivalent: (a) ( X × X, f × f ) is point-transitive and (b) ( ( F ( X ), d 0 ) is point-transitive.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. In this context, we consider the Zadeh’s extension f ^ of f to F ( X ), the family of all normal fuzzy sets on X, i.e., the hyperspace F ( X ) of all upper semicontinuous fuzzy sets on X with compact supports and non-empty levels and we endow F ( X ) with different metrics: the supremum metric d ∞, the Skorokhod metric d 0, the sendograph metric d S and the endograph metric d E. Given a metric space ( X, d ), we deal with a classical problem in the theory of hyperspaces: how some important dynamical properties (namely, weakly mixing, transitivity and point-transitivity) between a discrete dynamical system f : ( X, d ) → ( X, d ) and its natural extension to the hyperspace are related.