Baire-Type Properties in Metrizable c0(Ω, X)

Ferrando and Lüdkovsky proved that for a non-empty set Ω and a normed space X, the normed space c0(Ω,X) is barrelled, ultrabornological, or unordered Baire-like if and only if X is, respectively, barrelled, ultrabornological, or unordered Baire-like. When X is a metrizable locally convex space, with an increasing sequence of semi-norms .n∈N defining its topology, then c0(Ω,X) is the metrizable locally convex space over the field K (of the real or complex numbers) of all functions f:Ω→X such that for each ε>0 and n∈N the set ω∈Ω:f(ω)n>ε is finite or empty, with the topology defined by the semi-norms fn=supf(ω)n:ω∈Ω, n∈N. Kąkol, López-Pellicer and Moll-López also proved that the metrizable space c0(Ω,X) is quasi barrelled, barrelled, ultrabornological, bornological, unordered Baire-like, totally barrelled, and barrelled of class p if and only if X is, respectively, quasi barrelled, barrelled, ultrabornological, bornological, unordered Baire-like, totally barrelled, and barrelled of class p. The main result of this paper is that the metrizable c0(Ω,X) is baireled if and only if X is baireled, and its proof is divided in several lemmas, with the aim of making it easier to read. An application of this result to closed graph theorem, and two open problems are also presented.

Banach Spaces

The first four sections of this chapter form its core and include classical topics such as bounded linear transformations, the open mapping theorem, the closed graph theorem, the uniform boundedness principle, and the Hahn-Banach theorem. The chapter includes a good number of applications of the four fundamental theorems of functional analysis. Sections 6.5 and 6.6 provide a good account of the properties of the spectrum and adjoint operators on Banach spaces. They may be largely bypassed, since the treatment of the corresponding topics for operators on Hilbert spaces in chapter 7 is self-contained. The section on weak topologies is more advanced and may be omitted if a brief introduction is the goal. The chapter is enriched by such topics as the best polynomial approximation, the Hilbert cube, Gelfand’s theorem, Schauder bases, complemented subspaces, and the Banach-Alaoglu theorem.

Fuzzy Open Mapping and Fuzzy Closed Graph Theorems in Fuzzy Length Space

The theory of fuzzy set includes many aspects that regard important and significant in different fields of science and engineering in addition to there applications. Fuzzy metric and fuzzy normed spaces are essential structures in the fuzzy set theory. The concept of fuzzy length space has been given analogously and the properties of this space are studied few years ago. In this work, the definition of a fuzzy open linear operator is presented for the first time and the fuzzy Barise theorem is established to prove the fuzzy open mapping theorem in a fuzzy length space. Finally, the definition of a fuzzy closed linear operator on fuzzy length space is introduced to prove the fuzzy closed graph theorem.

Non-archimedean closed graph theorems

We consider linear maps T: X ? Y, where X and Y are polar local convex spaces over a complete non-archimedean field K. Recall that X is called polarly barrelled, if each weakly* bounded subset in the dual X0 is equicontinuous. If in this definition weakly* bounded subset is replaced by weakly* bounded sequence or sequence weakly* converging to zero, then X is said to be l?-barrelled or c0-barrelled, respectively. For each of these classes of locally convex spaces (as well as the class of spaces with weakly* sequentially complete dual) as domain class, the maximum class of range spaces for a closed graph theorem to hold is characterized. As consequences of these results, we obtain non-archimedean versions of some classical closed graph theorems. The final section deals with the necessity of the above-named barrelledness-like properties in closed graph theorems. Among others, counterparts of the classical theorems of Mahowald and Kalton are proved.

A simple proof of the closed graph theorem

<p>Assume that <em>A</em> is a closed linear operator defined on all of a Hilbert space <em>H</em>. Then, <em>A</em> is bounded. This classical theorem is proved on the basis of uniform boundedness principle. The proof is easily extended to Banach spaces.</p>

Fuzzy Continuous Mappings in Fuzzy Normed Linear Spaces

In this paper we continue the study of fuzzy continuous mappings in fuzzy normed linear spaces initiated by T. Bag and S.K. Samanta, as well as by I. Sadeqi and F.S. Kia, in a more general settings. Firstly, we introduce the notion of uniformly fuzzy continuous mapping and we establish the uniform continuity theorem in fuzzy settings. Furthermore, the concept of fuzzy Lipschitzian mapping is introduced and a fuzzy version for Banach’s contraction principle is obtained. Finally, a special attention is given to various characterizations of fuzzy continuous linear operators. Based on our results, classical principles of functional analysis (such as the uniform boundedness principle, the open mapping theorem and the closed graph theorem) can be extended in a more general fuzzy context.