
TL;DR
This paper introduces the concept of multi-normed spaces, generalizing normed spaces by considering sequences of norms on product spaces, and explores their properties, examples, and connections to operator theory.
Contribution
It defines and characterizes multi-normed spaces, linking them to tensor products, absolutely summing operators, and Banach lattice theory, with numerous examples and applications.
Findings
Multi-norms measure geometric features of normed spaces.
Established connections between multi-normed spaces and absolutely summing operators.
Developed a theory of multi-bounded operators and orthogonal decompositions.
Abstract
We modify the very well known theory of normed spaces within functional analysis by considering a sequence of norms, where is defined on the product space for each . Our theory is analogous to, but distinct from, an existing theory of `operator spaces'; it is designed to relate to general spaces for , and in particular to -spaces, rather than to -spaces. After recalling in Chapter 1 some results in functional analysis, especially in Banach space, Hilbert space, Banach algebra, and Banach lattice theory that we shall use, we shall present in Chapter 2 our axiomatic definition of a `multi-normed space' , where is a normed space. Several different, equivalent, characterizations of multi-normed spaces are given, some involving the theory of tensor products;…
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