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UDC 515.12 We show that a continuous open linear operator preserves the completeness and barreledness in locally convex cones. Specially, we prove some relations between an open linear operator and its adjoint in uc-cones (locally convex cones which their convex quasi-uniform structures are generated by one element).

Let $(U, R)$ be an approximation space with $U$ being non-empty set and $R$ being an equivalence relation on $U$, and let $\overline{G}$ and $\underline{G}$ be the upper approximation and the lower approximation of subset $G$ of $U$. A topological rough group $G$ is a rough group $G=(\underline{G}, \overline{G})$ endowed with a topology, which is induced from the upper approximation space $\overline{G}$, such that the product mapping $f: G\times G\rightarrow \overline{G}$ and the inverse mapping are continuous. In the class of topological rough groups, the relations of some separation axioms are obtained, some basic properties of the neighborhoods of the rough identity element and topological rough subgroups are investigated. In particular, some examples of topological rough groups are provided to clarify some facts about topological rough groups. Moreover, the version of open mapping theorem in the class of topological rough group is obtained. Further, some interesting open questions are posed., 19 pages

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.

Στη συγκεκριμένη διατριβή εξετάζουμε μερικές εφαρμογές του θεωρήματος Baire στη Μιγαδική ανάλυση, τόσο στη μία όσο και σε πολλές μιγαδικές μεταβλητές. Τα αποτελέσματα μας σχετίζονται με την έννοια της υπερκυκλικότητας, των πουθενά παραγωγίσιμων (μιγαδικών) συναρτήσεων, των προσεγγιστών Pade, όπως επίσης και με τις καθολικές σειρές Laurent.

A version of the classical Klee-And\^o Theorem states the following: For every Banach space $X$, ordered by a closed generating cone $C\subseteq X$, there exists some $\alpha>0$ so that, for every $x\in X$, there exist $x^{\pm}\in C$ so that $x=x^{+}-x^{-}$ and $\|x^{+}\|+\|x^{-}\|\leq\alpha\|x\|$. The conclusion of the Klee-And\^o Theorem is what is known as a conormality property. We prove stronger and somewhat more general versions of the Klee-And\^o Theorem for both conormality and coadditivity (a property that is intimately related to conormality). A corollary to our result shows that the functions $x\mapsto x^{\pm}$, as above, may be chosen to be bounded, continuous, and positively homogeneous, with a similar conclusion yielded for coadditivity. Furthermore, we show that the Klee-And\^o Theorem generalizes beyond ordered Banach spaces to Banach spaces endowed with arbitrary collections of cones. Proofs of our Klee-And\^o Theorems are achieved through an Open Mapping Theorem for cone-valued multi-functions/correspondences. We very briefly discuss a potential further strengthening of The Klee-And\^o Theorem beyond what is proven in this paper, and motivate a conjecture that there exists a Banach space $X$, ordered by a closed generating cone $C\subseteq X$, for which there exist no Lipschitz functions $(\cdot)^{\pm}:X\to C$ satisfying $x=x^{+}-x^{-}$ for all $x\in X$., Comment: Major rewrite. Large parts were removed which a referee pointed out can be proven through much easier methods

In this paper, we address a fundamental question about the nature of bilinear operators first posed by Walter Rudin. While it has been known that there is not an open mapping theorem for bilinear operators in general, we show that bilinear operators enjoy an open mapping theorem, when the range has dimension three or less, and we address the simple, but crucial property that differentiates bilinear operators from their linear counterparts. We also give an example which shows that polynomials, in general do not enjoy an open mapping theorem, even when the range has dimension three.

We discuss some properties of linear functionals on topological hyperbolic and topological bicomplex modules. The hyperbolic and bicomplex analogues of the uniform boundedness principle, the open mapping theorem, the closed graph theorem and the Hahn Banach separation theorem are proved.

We prove an open mapping theorem for the topological spaces dual to finitely presented Heyting algebras. This yields in particular a short, self-contained semantic proof of the uniform interpolation theorem for intuitionistic propositional logic, first proved by Pitts in 1992. Our proof is based on the methods of Ghilardi & Zawadowski. However, our proof does not require sheaves nor games, only basic duality theory for Heyting algebras., 8 pages. Minor changes in presentation. To appear in Topology and its Applications

In this paper we undertake to examine how algebra, its tools and its methods, can be used to formulate the mathematics used in applications. We give particular attention to the mathematics used in application to physics. We suggest that methods first proposed by Henry Siggins Leonard are well suited to such an examination.

Let $X,Y$ be Banach spaces. We define \begin{equation*}\alpha_{Y}(X)=\sup\{\vert T^{-1}\vert^{-1}:T:Y\rightarrow X\mbox{ is an isomorphism with }\vert T\vert \leq1\}.\end{equation*} If there is no isomorphism from $Y$ to $X$ , we set $\alpha_{Y}(X)=0$ , and ¶ \begin{equation*}\gamma_{Y}(X)=\sup\{\delta(T):T:X\rightarrow Y\mbox{ is asurjective operator with }\vert T\vert \leq1\},\end{equation*} where $\delta(T)=\sup\{\delta\gt 0:\delta B_{Y}\subseteq TB_{X}\}$ . If there is no surjective operator from $X$ onto $Y$ , we set $\gamma_{Y}(X)=0$ . We prove that for a separable space $X$ , $\alpha_{l_{1}}(X^{*})=\gamma_{c_{0}}(X)$ and $\alpha_{L_{1}}(X^{*})=\gamma_{C(\Delta)}(X)=\gamma_{C[0,1]}(X)$ .

We describe the multiplier algebra of the noncommutative Schwartz space. This multiplier algebra can be seen as the largest $^{*}$ -algebra of unbounded operators on a separable Hilbert space with the classical Schwartz space of rapidly decreasing functions as the domain. We show in particular that it is neither a $\mathcal{Q}$ -algebra nor $m$ -convex. On the other hand, we prove that classical tools of functional analysis, for example, the closed graph theorem, the open mapping theorem, or the uniform boundedness principle, are still available.

We investigate the validity of the Mean Ergodic Theorem in symmetric Banach function spaces E . The assertion of that theorem always holds when E is separable, whereas the situation is more delicate when E is non-separable. To describe positive results in the latter setting, we use the connections with the theory of singular traces.

In 1955, M. A. Krasnosel’skii proved a fixed-point theorem for a single-valued map which is a completely continuous contraction (a hybrid theorem). Subsequently, his work was continued in various directions. In particular, it has stimulated the development of the theory of condensing maps (both single-valued and set-valued); the images of such maps are always compact. Various versions of hybrid theorems for set-valued maps with noncompact images have also been proved. The set-valued contraction in these versions was assumed to have closed images and the completely continuous perturbation, to be lower semicontinuous (in a certain sense). In this paper, a new hybrid fixed-point theorem is proved for any set-valued map which is the sum of a set-valued contraction and a compact set-valued map in the case where the compact set-valued perturbation is upper semicontinuous and pseudoacyclic. In conclusion, this hybrid theorem is used to study the solvability of operator inclusions for a new class of operators containing all surjective operators. The obtained result is applied to solve the solvability problem for a certain class of control systems determined by a singular differential equation with feedback.

The existence of mild solution and the constrained local controllability of a retarded boundary control system with nonlocal delay condition have been established. The theory of extrapolation spaces is applied to derive the mild solution. Then, the constrained local controllability is established using the generalized open mapping theorem. In the last section, application of the result is shown through examples of control systems represented by hyperbolic partial differential equations.

The Livsic Theorem for Holder continuous cocycles with values in Banach rings is proved. We consider a transitive homeomorphism ${\sigma :X\to X}$ that satisfies the Anosov Closing Lemma and a Holder continuous map ${a:X\to B^\times}$ from a compact metric space $X$ to the set of invertible elements of some Banach ring $B$. The map $a(x)$ is a coboundary with a Holder continuous transition function if and only if $a(\sigma^{n-1}p)\ldots a(\sigma p)a(p)$ is the identity for each periodic point $p=\sigma^n p$.

We study the notion of a non-Archimedean fuzzy anti-2-normed space over a non-Archimedean field and prove that Mazur-Ulam theorem holds under some conditions in non-Archimedean fuzzy anti-2-normed spaces.

In this paper a nonsmooth mean value theorem in asplund spaces, under convexity, using the properties of limiting subdifferentials is established. We research on a kind of mean value theorem and prove that this theorem for set-valued mappings under convexity of domein in banach spaces. This theorem is use full to establish new results in convex analysis.

We prove that if K and S are locally compact Hausdorff spaces and there exists a bijective coarse ( M , L ) -quasi-isometry T between the Banach spaces of real continuous functions C 0 ( K ) and C 0 ( S ) with M 2 , then K and S are homeomorphic. This nonlinear extension of Banach–Stone theorem (1933/1937) is in some sense optimal and improves some results of Amir (1965), Cambern (1967), Jarosz (1989), Dutrieux and Kalton (2005) and Gorak (2011). In the Lipschitz case, that is when L = 0 , we also improve the estimations of the distance of the map T from the isometries between the spaces C 0 ( K ) and C 0 ( S ) obtained by Gorak when K and S are compact spaces or not. As a consequence, we get a linear sharp refinement of the Amir–Cambern theorem.

We provide a new and significantly shorter optimality proof of recent quantified Tauberian theorems, both in the setting of vector-valued functions and of $C_0$-semigroups, and in fact our results are also more general than those currently available in the literature. Our approach relies on a novel application of the open mapping theorem., Comment: To appear in the Israel Journal of Mathematics

This paper summarizes the research work on functional space theory, including the infinite matrix method, characterization of $({op})$ type space,invariance, nonlinear open mapping theorem, nonlinear closed graph theorem, demi-linear equicontinuity theorem, demi-linear duality theory, anddemi-linear generalized function theory.

In this paper we prove some new conditions for Cauchy sequences by using the diameter of orbit in partial metric spaces. A fixed point theorem for Meir-Keeler type contractions in this space is established.

We introduce the notions of extended topological vector spaces and extended seminormed spaces, following the main ideas of extended normed spaces, which were introduced by G. Beer and J. Vanderwerff. We provide a topological study of such structures, giving a unifying theory with main applications in the study of spaces of continuous functions. We also generalize classical results of functional analysis, as open mapping theorem and closed graph theorem.

Order unit normed linear spaces are a special type of regularly ordered normed linear spaces and therefore the first section is a short collection of the fundamental results on this type of normed linear spaces. The connection between order unit normed linear spaces and base normed linear spaces within the category of regularly ordered normed linear spaces is described in Section 2, and Section 3 at last, contains the results on Banach limits in an arbitrary order unit normed linear space. It is shown that the original results on Banach limits are valid for a greater range.

This work presents an Ovsyannikov type theorem for an autonomous abstract Cauchy problem in a scale of decreasing Banach spaces, which in addition to existence and uniqueness of solution provides an estimate about the analytic lifespan of the solution. Then, using this theorem it studies the Cauchy problem for Camassa–Holm type equations and systems with initial data in spaces of analytic functions on both the circle and the line, which is the main goal of this paper. Finally, it studies the continuity of the data-to-solution map in spaces of analytic functions.

The formal balls in fuzzy normed space X (characterized by closed balls in X) are ordered by reverse inclusion depending on the concept of level sets. The set of formal balls in a fuzzy normed space is called a fuzzy domain normed space denoted by BX. This set is directed complete partially ordered set (dcpo), its maximal elements are the suprema. A contraction mapping principle is defined on BX. Banach fixed point theorem is studied and proved on BX.

We present a computer formalization of metric spaces in the HOL Light theorem prover. Basic results of the theory of complete metric spaces are provided, including the Banach Fixed-Point Theorem, the Baire Category Theorem and the completeness of the space of continuous bounded functions. A decision procedure for a fragment of the elementary theory of metric spaces is also implemented. As an application, the Picard–Lindelof theorem on the existence of the solutions of ordinary differential equations is proved by using the well-known argument which appeals to the Banach theorem.

The abstract Cauchy problem on scales of Banach space was considered by many authors. The goal of this paper is to show that the choice of the space on scale is significant. We prove a theorem that the selection of the spaces in which the Cauchy problem ut − Δu = u|u|s with initial–boundary conditions is considered has an influence on the selection of index s. For the Cauchy problem connected with the heat equation we will study how the change of the base space influents the regularity of the solutions.

In this article, we deal with weak convergence on sequences in real normed spaces, and weak* convergence on sequences in dual spaces of real normed spaces. In the first section, we proved some topological properties of dual spaces of real normed spaces. We used these theorems for proofs of Section 3. In Section 2, we defined weak convergence and weak* convergence, and proved some properties. By RNS_Real Mizar functor, real normed spaces as real number spaces already defined in the article [18], we regarded sequences of real numbers as sequences of RNS_Real. So we proved the last theorem in this section using the theorem (8) from [25]. In Section 3, we defined weak sequential compactness of real normed spaces. We showed some lemmas for the proof and proved the theorem of weak sequential compactness of reflexive real Banach spaces. We referred to [36], [23], [24] and [3] in the formalization.

We study the completeness properties of several different group topologies for the additive group of real numbers, and we also compute the corresponding dual groups. We first present two metrizable connected group topologies on R with topologically isomorphic dual groups, one of which is noncomplete and arcwise connected and the other one is compact (therefore complete), but not arcwise connected. Using a theorem about T -sequences and adapting a result about weakened analytic groups, we then describe a method for obtaining Hausdorff group topologies R that are strictly weaker than the usual topology and are complete. They are not Baire, and consequently not metrizable.

It is well-known that the state space isomorphism theorem fails in infinite-dimensional Hilbert spaces: there exist minimal discrete-time systems (with Hilbert space state spaces) which have the same impulse response, but which are not isomorphic. We consider discrete-time systems on locally convex topological vector spaces which are Hausdorff and barrelled and show that in this setting the state space isomorphism theorem does hold. In contrast to earlier work on topological vector spaces, we consider a definition of minimality based on dilations and show how this necessitates certain definitions of controllability and observability.

In this paper, we prove a fixed point theorem for a self map in fuzzy metric space where the map satisfies a different condition.

In this paper, we study Baire property of a family of spaces which contains properly the family of all topological spaces and generalize the existing results. Also, we study the images and inverse images of such spaces.

We prove a closed graph theorem for Baire locally convex spaces (for Baire linear topological spaces) in the domain and weakly C-Suslin locally convex spaces (respectively, for C-Suslin linear topological spaces) in the range which improves some classic closed graph theorems and other, more recent, related results.

Classical Marchaud's theorem (1927) asserts that if f is a bounded function on [ a , b ] , k ∈ N , and the ( k + 1 ) th modulus of smoothness ω k + 1 ( f ; t ) is so small that η ( t ) = ∫ 0 t ω k + 1 ( f ; s ) s k + 1 d s + ∞ for t > 0 , then f ∈ C k ( ( a , b ) ) and f ( k ) is uniformly continuous with modulus cη for some c > 0 (i.e. in our terminology f is C k , c η -smooth). Using a known version of the converse of Taylor theorem we easily deduce Marchaud's theorem for functions on certain open connected subsets of Banach spaces from the classical one-dimensional version. In the case of a bounded subset of R n our result is more general than that of H. Johnen and K. Scherer (1973), which was proved by quite a different method. We also prove that if a locally bounded mapping between Banach spaces is C k , ω -smooth on every line, then it is C k , c ω -smooth for some c > 0 .

The Lebesgue type decomposition theorem and weak Radon-Nikodým theorem for fuzzy valued measures in separable Banach spaces are established.

We survey some aspects of the classical prediction theory for stationary processes, in discrete time in Section 1, turning in Section 2 to continuous time, with particular reference to reproducing-kernel Hilbert spaces and the sampling theorem. We discuss the discrete-continuous theories of ARMA-CARMA, GARCH-COGARCH, and OPUC-COPUC in Section 3. We compare the various models treated in Section 4 by how well they model volatility, in particular volatility clustering. We discuss the infinite-dimensional case in Section 5, and turn briefly to applications in Section 6.

Tchakaloff’s theorem gives a quadrature formula for polynomials of a given degree with respect to a compactly supported positive measure which is absolutely continuous with respect to Lebesgue measure. We study the validity of two possible analogues of Tchakaloff’s theorem in an infinite-dimensional Banach space E E : a weak form valid when E E has a Schauder basis, and a stronger form requiring conditions on the support of the measure as well as on the space E E .

New results on the Baire product problem are presented. It is shown that an arbitrary product of almost locally ccc Baire spaces is Baire; moreover, the product of a Baire space and a 1st countable space which is β-unfavorable in the strong Choquet game is Baire.

This work studies slice functions over finite-dimensional division algebras. Their zero sets are studied in detail along with their multiplicative inverses, for which some unexpected phenomena are discovered. The results are applied to prove some useful properties of the subclass of slice regular functions, previously known only over quaternions. Firstly, they are applied to derive from the maximum modulus principle a version of the minimum modulus principle, which is in turn applied to prove the open mapping theorem. Secondly, they are applied to prove, in the context of the classification of singularities, the counterpart of the Casorati-Weierstrass theorem., Comment: 24 pages, published online in Proc. Roy. Soc. Edinburgh Sect. A

This paper generalizes the Aleksandrov problem, the Mazur-Ulam theorem and Benz theorem on n-normed spaces. It proves that a one-distance preserving mapping is an n- isometry if and only if it has the zero-distance preserving property, and two kinds of n-isometries on n-normed spaces are equivalent.

Under some additional assumptions on an unbounded sequence of operators and the geometry of the spaces, it is shown that, in the classical Banach-Steinhaus resonance theorem, the set of divergence contains an infinite-dimensional space, excluding zero.

Summary In this article, we deal with dual spaces and the Hahn-Banach Theorem. At the first, we defined dual spaces of real linear spaces and proved related basic properties. Next, we defined dual spaces of real normed spaces. We formed the definitions based on dual spaces of real linear spaces. In addition, we proved properties of the norm about elements of dual spaces. For the proof we referred to descriptions in the article [21]. Finally, applying theorems of the second section, we proved the Hahn-Banach extension theorem in real normed spaces. We have used extensively used [17].