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"Functions, Continuous."
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Some results and open questions on spaceability in function spaces
A subset MM of a topological vector space XX is called lineable (respectively, spaceable) in XX if there exists an infinite dimensional linear space (respectively, an infinite dimensional closed linear space) Y⊂M∪{0}Y \\subset M\\cup \\{0\\}. In this article we prove that, for every infinite dimensional closed subspace XX of C[0,1]\\mathcal {C}[0,1], the set of functions in XX having infinitely many zeros in [0,1][0,1] is spaceable in XX. We discuss problems related to these concepts for certain subsets of some important classes of Banach spaces (such as C[0,1]\\mathcal {C}[0,1] or Müntz spaces). We also propose several open questions in the field and study the properties of a new concept that we call the oscillating spectrum of subspaces of C[0,1]\\mathcal {C}[0,1], as well as oscillating and annulling properties of subspaces of C[0,1]\\mathcal {C}[0,1].
Journal Article
On Strong Porosity of Some Families of Functions
The paper deals with the strong porosity of some families of real functions continuous with respect to a given topology or -continuous (i.e., continuous with respect to some special family of sets of the real line). Particularly, porosity of those families is investigated in space of the Baire 1 functions or in the space of the Baire 1 and Darboux functions.
Journal Article
Ascoli-Arzelà Theorem (Metric Space Version)
We formulate and prove in Mizar the Ascoli-Arzelà’s theorem, which gives necessary and sufficient conditions for a collection of continuous functions to be compact. We use the metric space setting, and the notions of equicontinuousness and equiboundedness of a set of continuous functions are utilized.
Journal Article
Some New Notions of Continuity in Generalized Primal Topological Space
This study analyzes the characteristics and functioning of Sg∗-functions, Sg∗-homeomorphisms, and Sg∗#-homeomorphisms in generalized topological spaces (GTS). A few important points to emphasize are Sg∗-continuous functions, Sg∗-irresolute functions, perfectly Sg∗-continuous, and strongly Sg∗-continuous functions in GTS and generalized primal topological spaces (GPTS). Some specific kinds of Sg∗ functions, such as Sg∗-open mappings and Sg∗-closed mappings, are discussed. We also analyze the GPTS, providing a thorough look at the way these functions work in this specific context. The goal here is to emphasize the concrete implications of Sg∗ functions and to further the theoretical understanding of them by merging different viewpoints. This work advances the area of topological research by providing new perspectives on the behavior of Sg∗ functions and their applicability in various topological settings. The outcomes reported here contribute to our theoretical understanding and establish a foundation for further research.
Journal Article
Embeddings of Decomposition Spaces
Many smoothness spaces in harmonic analysis are decomposition spaces. In this paper we ask: Given two such spaces, is there an
embedding between the two?
A decomposition space
We establish readily verifiable criteria which ensure the
existence of a continuous inclusion (“an embedding”)
In a nutshell, in order to apply the embedding results presented in this
article, no knowledge of Fourier analysis is required; instead, one only has to study the geometric properties of the involved
coverings, so that one can decide the finiteness of certain sequence space norms defined in terms of the coverings.
These
sufficient criteria are quite sharp: For almost arbitrary coverings and certain ranges of
We also prove a
The resulting embedding theory is illustrated by applications
to
eBook
Symmetric Markov processes, time change, and boundary theory
This book gives a comprehensive and self-contained introduction to the theory of symmetric Markov processes and symmetric quasi-regular Dirichlet forms. In a detailed and accessible manner, Zhen-Qing Chen and Masatoshi Fukushima cover the essential elements and applications of the theory of symmetric Markov processes, including recurrence/transience criteria, probabilistic potential theory, additive functional theory, and time change theory. The authors develop the theory in a general framework of symmetric quasi-regular Dirichlet forms in a unified manner with that of regular Dirichlet forms, emphasizing the role of extended Dirichlet spaces and the rich interplay between the probabilistic and analytic aspects of the theory. Chen and Fukushima then address the latest advances in the theory, presented here for the first time in any book. Topics include the characterization of time-changed Markov processes in terms of Douglas integrals and a systematic account of reflected Dirichlet spaces, and the important roles such advances play in the boundary theory of symmetric Markov processes.
This volume is an ideal resource for researchers and practitioners, and can also serve as a textbook for advanced graduate students. It includes examples, appendixes, and exercises with solutions.
eBook
On Micro-α-Open Sets and Micro-α- Continuous Functions in Micro Topological Spaces
The principle point ofithis paper is to present new type of topologicalispaces are introduced by nano topology and so on Nano topology introduced by Thivagar using this nano topology we introduced micro topology and also introduce new type of open sets namely of Micro-α-open sets furthermore, a portion of their properties are researched. As application to Micro-α-open sets we introduce Micro-α-continuous functions also, get a portion of their essential properties
Journal Article
SUZUKI-WARDOWSKI TYPE FIXED POINT THEOREMS FORα-GF-CONTRACTIONS
Recently, Wardowski [Fixed Point Theory Appl. 2012:94, 2012] introduced and studied a new contraction called F-contraction to prove a fixed point result as a generalization of the Banach contraction principle. Abbas et al. [2] further generalized the concept of F-contraction and proved certain fixed and common fixed point results. In this paper, we introduce anα-GF-contraction with respect to a general family of functionsGand establish Wardowski type fixed point results in metric and ordered metric spaces. As an application of our results we deduce Suzuki type fixed point results for GF-contractions. We also derive certain fixed and periodic point results for orbitally continuous generalized F-contractions. Moreover, we discuss some illustrative examples to highlight the realized improvements.
2010Mathematics Subject Classification: 46N40, 47H10, 54H25, 46T99.
Key words and phrases: Fixed point,α-GF-contraction,α-η-Continous function, Orbitally continuous function.
Journal Article
Some Properties in Grill–Topological Open and Closed Sets
The aim of this research is to study the modern sort of open sets that is said -g-closed. Some functions by this notion was explained and the relationships among them like continuous function g-continuous function strongly- g-continuous function and g-irresolute function.
Journal Article