This new volume shows how it is possible to further develop and essentially extend the theory of operators in infinite-dimensional vector spaces, which plays an important role in mathematics, physics, information theory, and control theory. The book describes new mathematical structures, such as hypernorms, hyperseminorms, hypermetrics, semitopological vector spaces, hypernormed vector spaces, and hyperseminormed vector spaces. It develops mathematical tools for the further development of functional analysis and broadening of its applications.
Exploration of semitopological vector spaces, hypernormed vector spaces, hyperseminormed vector spaces, and hypermetric vector spaces is the main topic of this book. A new direction in functional analysis, called quantum functional analysis, has been developed based on polinormed and multinormed vector spaces and linear algebras. At the same time, normed vector spaces and topological vector spaces play an important role in physics and in control theory.
To make this book comprehendible for the reader and more suitable for students with some basic knowledge in mathematics, denotations and definitions of the main mathematical concepts and structures used in the book are included in the appendix, making the book useful for enhancing traditional courses of calculus for undergraduates, as well as for separate courses for graduate students. The material of Semitopological Vector Spaces: Hypernorms, Hyperseminorms and Operators is closely related to what is taught at colleges and universities. It is possible to use a definite number of statements from the book as exercises for students because their proofs are not given in the book but left for the reader.
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More than 20 years ago, the author introduced an extension R! of the real numbers whose
elements he called real hypernumbers. R! is a real ordered vector space, it contains arbitrar-
ily large" elements (i.e., elements which are larger than each real number) but no in nitely
small" positive elements. A few years ago, he published a book in which he showed that real
hypernumbers enable one to di
erentiate and integrate any real-valued function and studied
the outcoming properties [M. Burgin, Hypernumbers and extrafunctions. Extending the clas-
sical calculus. New York, NY: Springer (2012; Zbl 1253.46050)]. The present book now mainly
deals with hypernormed spaces: a mapping k < k : E ! R! de ned on a real vector space E
is called hyper(semi)norm and (E; k < k) a hyper(semi)normed space if it satis es the usual
conditions for a (semi)norm. Using the norm balls with positive real radius, a hyperseminorm
induces a topology which is a group topology but, in general, not a vector space topology, only
the multiplication with a xed real number is continuous (in other words, it is a topological
vector space over the reals, endowed with the discrete topology).
After the introduction of generalized P-hypernumbers, a generalization of real hypernumbers,
the author studies hyperfunctions and extrafunctions which allow him to introduce generalized
distributions.
Semitopological vector spaces are, with few exceptions, only discussed in one chapter. They
have been studied under the name topological vector groups" by, e.g., Raikov, Kenderov and
Lurje, with some remarkable generalizations of classical results. As there are few far-reaching
results for general topological vector spaces, the same is even more true for semitopological
vector spaces. It is shown that a family of hyperseminorms induces a semitopology which is
locally convex.
One of the main goals of the book is the study of objects or concepts which are approxima-
tively" equal. Since this approximation" is usually expressed in terms of a hyperseminorm,
the author studies these in detail in Chapter 3. This enables him to study fuzzy continuity,
approximative linear operators and bounded mappings, as well as their mutual relations. (Note
that the de nition of fuzzy continuity here is di
erent from the classical one.) This is done in
the last chapters of the book.
Most of the results in this book are elementary. It contains many minor and some major
mistakes. For instance, Proposition 3.4 claims that the product of norms is a norm, which is
evidently wrong. The proofs of Proposition 5.1 (b), (c), which characterizes compact Hausdor
semitopological vector spaces and claims that each nite-dimensional subspace of a Hausdor
semitopological vector space is closed, respectively, use the continuity of the scalar multiplica-
tion in the rst variable and are therefore wrong. Also, there are inconsistencies, so r-continuity
as de ned in De nition 7.3 is di
erent from "-continuity as de ned in De nition 7.10. Also,
Theorems 8.1 and 8.2 are missing, whereas De nitions 3.41 of a hyperquasimetric and 3.42 of
a hypersemimetric de ne the same object.
The author states that this book belongs to physical mathematics, but there are no examples
from this area as would have been desirable from the reviewer's point of view.
Heinz-Peter Butzmann (Mannheim)
