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Register a new userDirect and inverse limits of normed modules
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Authors
Enrico Pasqualetto
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The aim of this note is to study existence and main properties of direct and inverse limits in the category of normed $L^0$-modules (in the sense of Gigli) over a metric measure space.
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We modify the very well known theory of normed spaces $(E, orm)$ within functional analysis by considering a sequence $( orm_n : ninN)$ of norms, where $ orm_n$ is defined on the product space $E^n$ for each $ninN$. Our theory is analogous to, but distinct from, an existing theory of `operator spaces; it is designed to relate to general spaces $L^p$ for $pin [1,infty]$, and in particular to $L^1$-spaces, rather than to $L^2$-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 $((E^n, orm_n) : nin N)$, where $(E, orm)$ is a normed space. Several different, equivalent, characterizations of multi-normed spaces are given, some involving the theory of tensor products; key examples of multi-norms are the minimum and maximum multi-norm based on a given space. Multi-norms measure `geometrical features of normed spaces, in particular by considering their `rate of growth. There is a strong connection between multi-normed spaces and the theory of absolutely summing operators. A substantial number of examples of multi-norms will be presented. Following the pattern of standard presentations of the foundations of functional analysis, we consider generalizations to `multi-topological linear spaces through `multi-null sequences, and to `multi-bounded linear operators, which are exactly the `multi-continuous operators. We define a new Banach space ${mathcal M}(E,F)$ of multi-bounded operators, and show that it generalizes well-known spaces, especially in the theory of Banach lattices. We conclude with a theory of `orthogonal decompositions of a normed space with respect to a multi-norm, and apply this to construct a `multi-dual space.
Let X, Y be asymmetric normed spaces and Lc(X, Y) the convex cone of all linear continuous operators from X to Y. It is known that in general, Lc(X, Y) is not a vector space. The aim of this note is to prove, using the Baire category theorem, that if Lc(X, Y) is a vector space for some asymmetric normed space Y , then X is isomorphic to its associated normed space (the converse is true for every asymmetric normed space Y and is easy to establish). For this, we introduce an index of symmetry of the space X denoted c(X) $in$ [0, 1] and we give the link between the index c(X) and the fact that Lc(X, Y) is in turn an asymmetric normed space for every asymmetric normed space Y. Our study leads to a topological classification of asymmetric normed spaces.
We clarify the relation between inverse systems, the Radon-Nikodym property, the Asymptotic Norming Property of James-Ho, and the GFDA spaces introduced in our earlier paper on differentiability of Lipschitz maps into Banach spaces.
The concept of fuzzy soft set was introduced for the first time by Maji et al. in 2002, and was considered sharply from applicable aspects to theoretical aspects by a wide range of researchers. In this paper the concept of fuzzy soft norm over fuzzy soft spaces has been considered and some properties of fuzzy soft normed spaces are studied. We also study the fuzzy soft topology over a crisp set by using the fuzzy soft subsets of it and the relationship between fuzzy soft topology and general topology is investigated. Fuzzy soft linear operator over fuzzy soft spaces is introduced and continuity of such operators is considered.
Given a locally compact abelian group $G$ and a closed subgroup $Lambda$ in $Gtimeswidehat{G}$, Rieffel associated to $Lambda$ a Hilbert $C^*$-module $mathcal{E}$, known as a Heisenberg module. He proved that $mathcal{E}$ is an equivalence bimodule between the twisted group $C^*$-algebra $C^*(Lambda,textsf{c})$ and $C^*(Lambda^circ,bar{textsf{c}})$, where $Lambda^{circ}$ denotes the adjoint subgroup of $Lambda$. Our main goal is to study Heisenberg modules using tools from time-frequency analysis and pointing out that Heisenberg modules provide the natural setting of the duality theory of Gabor systems. More concretely, we show that the Feichtinger algebra ${textbf{S}}_{0}(G)$ is an equivalence bimodule between the Banach subalgebras ${textbf{S}}_{0}(Lambda,textsf{c})$ and ${textbf{S}}_{0}(Lambda^{circ},bar{textsf{c}})$ of $C^*(Lambda,textsf{c})$ and $C^*(Lambda^circ,bar{textsf{c}})$, respectively. Further, we prove that ${textbf{S}}_{0}(G)$ is finitely generated and projective exactly for co-compact closed subgroups $Lambda$. In this case the generators $g_1,ldots,g_n$ of the left ${textbf{S}}_{0}(Lambda)$-module ${textbf{S}}_{0}(G)$ are the Gabor atoms of a multi-window Gabor frame for $L^2(G)$. We prove that this is equivalent to $g_1,ldots,g_n$ being a Gabor super frame for the closed subspace generated by the Gabor system for $Lambda^{circ}$. This duality principle is of independent interest and is also studied for infinitely many Gabor atoms. We also show that for any non-rational lattice $Lambda$ in $mathbb{R}^{2m}$ with volume ${s}(Lambda)<1$ there exists a Gabor frame generated by a single atom in ${textbf{S}}_{0}(mathbb{R}^m)$.
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