It is well known that every (real or complex) normed linear space $L$ is isometrically embeddable into $C(X)$ for some compact Hausdorff space $X$. Here $X$ is the closed unit ball of $L^*$ (the set of all continuous scalar-valued linear mappings on $L$) endowed with the weak$^*$ topology, which is compact by the Banach-Alaoglu theorem. We prove that the compact Hausdorff space $X$ can indeed be chosen to be the Stone-Cech compactification of $L^*\setminus\{0\}$, where $L^*\setminus\{0\}$ is endowed with the supremum norm topology.
Let $(X,\left\Vert \cdot \right\Vert )$ be a real normed space of dimension $N\in \mathbb{N}$ with a basis $(e_{i})_{1}^{N}$ such that the norm is invariant under coordinate permutations. Assume for simplicity that the basis constant is at most $2$. Consider any $n\in \mathbb{N}$ and $0<\varepsilon <1/4$ such that $n\leq c(\log \varepsilon ^{-1})^{-1}\log N$. We provide an explicit construction of a matrix that generates a $(1+\varepsilon )$ embedding of $\ell _{2}^{n}$ into $X$.
Given a normed cone $(X,p)$ and a subcone $Y,$ we construct and study the quotient normed cone $(X/Y,\tilde{p})$ generated by $Y$. In particular we characterize the bicompleteness of $(X/Y,\tilde{p})$ in terms of the bicompleteness of $(X,p),$ and prove that the dual quotient cone $((X/Y)^{*},\|\cdot \|_{\tilde{p},u})$ can be identified as a distinguished subcone of the dual cone $(X^{*},\|\cdot \|_{p,u})$. Furthermore, some parts of the theory are presented in the general setting of the space $CL(X,Y)$ of all continuous linear mappings from a normed cone $(X,p)$ to a normed cone $(Y,q),$ exte…
The aim of this paper is to present a survey of some recent results obtained in the study of spaces with asymmetric norm. The presentation follows the ideas from the theory of normed spaces (topology, continuous linear operators, continuous linear functionals, duality, geometry of asymmetric normed spaces, compact operators) emphasizing similarities as well as differences with respect to the classical theory. The main difference comes form the fact that the dual of an asymmetric normed space $X$ is not a linear space, but merely a convex cone in the space of all linear functionals on $X.$ Due …
This article generalize the classical Goldstine-Weston theorem on normed spaces to one on random normed modules: the image of a random normed module $(E,\|\cdot\|)$ under the random natural embedding $J$ is dense in its double random conjugate space $E^{**}$ with respect to the $(ε,λ)$ weak star topology; and $J(E)$ is also dense in $E^{**}$ with respect to the locally $L^{0}$-convex weak star topology if $E$ has the countable concatenation property.
Some properties of the (normed) dual Hom-functor $D$ and its iterations $D^n$ are exhibited. For instance: $D$ turns every canonical embedding (in the second dual space) into a retraction (of the third dual onto the first one); $D$ rises the countably infinite (algebraic) dimension only; $D$ does not change the finite quotient shape type. By means of that, the finite quotient shape classification of normed vectorial spaces is completely solved. As a consequence, two extension type theorems are derived.
In this paper, we study a part of approximation theory that presents the conditions under which a closed set in a normed linear space is proximinal or Chebyshev.
In this paper we discuss some properties of resolvents of an accretive operator in linear 2-normed spaces, focusing on the concept of contraction mapping and the unique fixed point of contraction mappings in linear 2- normed spaces. Also, we establish the existence of solution of strong accretive operator equation in linear 2-normed spaces.
In this paper, we shall investigate and analyse a new study on the best simultaneous approximation in the context of linear 2-normed spaces inspired by Elumalai and his coworkers in Elumalai. The basis of this investigation is to extend and refinement the definition of the classical aproximation, best approximation and some related concepts to linear 2-normed spaces.
Based on ideas of R.W. Cross, a simplified proof is presented of the density invariance of certain operational quantities associated with bounded linear operators in normed vector spaces.
The concept of b-linear functional and its different types of continuity in linear n-normed space are presented and some of their properties are being established. We derive the Uniform Boundedness Principle and Hahn-Banach extension Theorem with the help of bounded b-linear functionals in the case of linear n-normed spaces and discuss some examples and applications. Finally, we present the concept of weak*convergence for the sequence of bounded b-linear functionals in linear n-normed space.
In this paper, we will see that the Cartesian product of two 2-Banach spaces is also 2-Banach space and discuss some properties of closed linear operator in linear 2-normed space. We also describe the concept of different types of continuity of b-linear functional and derive the Uniform Boundedness Principle and Hahn-Banach extension theorem for b-linear functionals in the case of linear 2-normed spaces. We also introduce the notion of weak*convergence for the sequence of bounded b-linear functionals relative to linear 2-normed space.
We introduce the notion of approximate smoothness in a normed linear space. We characterize this property and show the connections between smoothness and approximate smoothness for some spaces. As an application, we consider in particular the Birkhoff-James orthogonality and its right-additivity under the assumption of approximate smoothness.
We consider linear narrow operators on lattice-normed spaces. We prove that, under mild assumptions, every finite rank linear operator is strictly narrow (before it was known that such operators are narrow). Then we show that every dominated, order continuous linear operator from a lattice-normed space over atomless vector lattice to an atomic lattice-normed space is order narrow.
We give the symmetric version of five lemmas which are essential for the theory of tensor products (and norms). These are: the approximation, extension, embedding, density and local technique lemmas. Some application of these tools to the metric theory of symmetric tensor products and to the theory of polynomials ideals are given.
In this paper we introduce the hypo-q-norms on a Cartesian product of normed linear spaces. A representation of these norms in terms of bounded linear functionals of norm less than one, the equivalence with the q-norms on a Cartesian product and some reverse inequalities obtained via the scalar Shisha-Mond, Birnacki et al. and other Gruss type inequalities are also given.
We obtain a complete characterization of the norm attainment set of a bounded linear functional on a normed space, in terms of a semi-inner-product defined on the space. Motivated by this result, we further apply the concept of semi-inner-products to obtain a complete characterization of the norm attainment set of a bounded linear operator between any two real normed spaces. In particular, this answers an open question raised recently in [Sain, D., \textit{On the norm attainment set of a bounded linear operator}, J. Math. Anal. Appl., \textbf{457} (2018), 67-76.]. Our results illustrate the ap…
Let $B_Y$ denote the unit ball of a normed linear space $Y$. A symmetric, bounded, closed, convex set $A$ in a finite dimensional normed linear space $X$ is called a {\it sufficient enlargement} for $X$ if, for an arbitrary isometric embedding of $X$ into a Banach space $Y$, there exists a linear projection $P:Y\to X$ such that $P(B_Y)\subset A$. The main results of the paper: {\bf (1)} Each minimal-volume sufficient enlargement is linearly equivalent to a zonotope spanned by multiples of columns of a totally unimodular matrix. {\bf (2)} If a finite dimensional normed linear space has a minima…
We give an affirmative answer to a question asked by Faghih-Ahmadi and Hedayatian regarding supercyclic vectors. We show that if $\mathcal X$ is an infinite-dimensional normed linear space and $T$ is a supercyclic operator on $\mathcal X$, then for any supercyclic vector $x$ for $T$, there exists a strictly increasing sequence $(n_k)_k$ of positive integers such that the closed linear span of the set $\{T^{n_k}x: k\ge 1\}$ is not the whole $\mathcal X$.
In this paper we introduce the notion of D-valued 2-norm on hy- perbolic or D-valued modules. Further, we define D-linear 2-functional on these modules and consider some of their properties. We also establish the Hahn- Banach type extension theorem for D-linear 2-functionals.
In this short note, we show by elementary computations that the notion of non-Archimedean fuzzy normed (and 2-normed) spaces is void. Namely, there are no strictly convex spaces at all --not even the zero-dimensional linear space. Before this, we also study the case of strictly convex non-Archimedean normed spaces; in this setting we see that the only nonzero linear space (defined over an arbitrary non-Archimedean field) that satisfies this property is the one-dimensional linear space over $\mathbb{Z}/3\mathbb{Z}$. Consequently, the results that have been proven for this class of spaces, like …
In this paper, we introduce the cone normed spaces and cone bounded linear mappings. Among other things, we prove the Baire category theorem and the Banach--Steinhaus theorem in cone normed spaces.
Let $H$ be a reflexive, dense, separable, infinite dimensional complex Hilbert space and let $B(H)$ be the algebra of all bounded linear operators on $H$. In this paper, we carry out characterizations of norm-attainable operators in normed spaces. We give conditions for norm-attainability of linear functionals in Banach spaces, non-power operators on $H$ and elementary operators. Lastly, we characterize a new notion of norm-attainability for power operators in normed spaces.
In this paper, we present a geometric form of the Hahn-Banach extension theorem for $L^{0}-$linear functions and prove that the geometric form is equivalent to the analytic form of the Hahn-Banach extension theorem. Further, we use the geometric form to give a new proof of a known basic strict separation theorem in random locally convex modules. Finally, using the basic strict separation theorem we establish the Goldstine-Weston theorem in random normed modules under the two kinds of topologies----the $(ε,λ)-$topology and the locally $L^{0}-$convex topology, and also provide a counterexample s…
The purpose of this paper is to introduce a new Kirk type iterative algorithm called Kirk multistep iteration and to study its convergence. We also prove some theorems related with the stability results for the Kirk-multistep and Kirk-SP iterative processes by employing certain contractive-like operators. Our results generalize and unify some other results in the literature.
The concept of Soft set theory was introduced by Molodtsov in the study [8]. Soft real numbers and properties were introduced inthe study [6] and soft normed space was defined in [11]. In this study, firstly we obtain a soft normed space by defining a soft norm on (Real numbers) which is called soft normed real space. By using this normed space we define the soft linear functional and investigate some of its properties. Secondly, we introduce soft dual space and soft dual operator and investigate their properties. Finally, we state and prove the theorem about representation of soft linear func…
In this paper, we work on the structure of soft linear spaces over a field K and investigate some of its properties. Here, we use the concept of the soft point which was introduced in [2,6]. We then introduce the soft norm in soft linear spaces. Finally, we examine the properties of this soft normed space and present some investigations about soft continuous operators in the space.
In this paper, we prove that if $E$ is a uniquely remotal subset of a real normed linear space $X$ such that $E$ has a Chebyshev center $c \in X$ and the farthest point map $F:X\rightarrow E$ restricted to $[c,F(c)]$ is partially statistically continuous at $c$, then $E$ is a singleton. We obtain a necessary condition on uniquely remotal subsets of uniformly rotund Banach spaces to be a singleton. Moreover, we show that there exists a remotal set $M$ having a Chebyshev center $c$ such that the farthest point map $F:\mathbb{R}\rightarrow M$ is not continuous at $c$ but is partially statisticall…
In this paper the weak topology on a normed space is studied from the viewpoint of infinite-dimensional topology. Besides the weak topology on a normed space $X$ (coinciding with the topology of uniform convergence on finite subsets of the dual space $X^*$), we consider the topology $c$ of uniform convergence on compact subsets of $X^*$. It is known that this topology coincides with the weak topology on bounded subsets of $X$, but unlike to the latter has much better topological properties (e.g., is stratifiable). We prove that for normed spaces $X,Y$ with separable duals the spaces $(X,weak)$…
Let $X$ and $Y$ be real normed spaces and $f \colon X\to Y$ a surjective mapping. Then $f$ satisfies $\{\|f(x)+f(y)\|, \|f(x)-f(y)\|\} = \{\|x+y\|, \|x-y\|\}$, $x,y\in X$, if and only if $f$ is phase equivalent to a surjective linear isometry, that is, $f=σU$, where $U \colon X\to Y$ is a surjective linear isometry and $σ\colon X\to \{-1,1\}$. This is a Wigner's type result for real normed spaces.
In this article, we study the notions of $n$-isometries in non-Archimedean $n$-normed spaces over linear ordered non-Archimedean fields, and prove the Mazur-Ulam theorem in the spaces. Furthermore, we obtain some properties for $n$-isometries in non-Archimedean $n$-normed spaces.
In this note we generalize the well-known Wigner's unitary-anti\-unitary theorem. For $X$ and $Y$ smooth normed spaces and $f:X\to Y$ a surjective mapping such that $|[f(x),f(y)]|=|[x,y]|$, $x,y\in X$, where $[\cdot,\cdot]$ is the unique semi-inner product, we show that $f$ is phase equivalent to either a linear or an anti-linear surjective isometry. When $X$ and $Y$ are smooth real normed spaces and $Y$ strictly convex, we show that Wigner's theorem is equivalent to $\{\|f(x)+f(y)\|,\|f(x)-f(y)\|\}=\{\|x+y\|,\|x-y\|\}$, $x,y\in X$.
Some additive reverses of the generalised triangle inequality in normed linear spaces are given. Applications for complex numbers are provided as well.
Let X be a normed linear space. We examine if every open, convex and unbounded subset of X is equal to the union of a family of open straight half lines. The answer is affirmative if and only if X is finite dimensional.
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 continuit…
In this paper, we apply the publication of Joung (2009) to derive a stability result for for the second order linear functional equation: $f(x) = pf(x-1)-qf(x-2)$ for all $x\in\mathbb R$, where $f$ is a mapping from $\mathbb R$ into the induced random space of any Banach space. By relaxing the lower bound assumption, we also generalize the result of Jung (2009) on arbitrary random normed spaces with the minimum $t$-norm. However, we need the monotonicity of the distribution in the lower bound assumption. By the properties of normal distributions, our main result can be applied.
In order to measure qualitative properties we introduce a notion of a type for arbitrary normed spaces which measures the worst possible growth of partial sums of sequences weakly converging to zero. The ideas can be traced back to Banach and Mazur who used this type to compare the so-called linear dimension of classical Banach spaces. As an application we compare the linear dimension and investigate isomorphy of some classical Banach spaces.
We show that every dominated linear operator from an Banach-Kantorovich space over atomless Dedekind complete vector lattice to a sequence Banach lattice $l_p(Γ)$ or $c_0(Γ)$ is narrow. As a conse- quence, we obtain that an atomless Banach lattice cannot have a finite dimensional decomposition of a certain kind. Finally we show that if a linear dominated operator T from lattice-normed space V to Banach- Kantorovich space W is order narrow then the same is its exact dominant $\ls T\rs$.
Two different notions of approximate Birkhoff-James orthogonality in nor\-med linear spaces have been introduced by Dragomir and Chmie\-liń\-ski. In the present paper we consider a global and a local approximate symmetry of the Birkhoff-James orthogonality related to each of the two definitions. We prove that the considered orthogonality is approximately symmetric in the sense of Dragomir in all finite-dimensional Banach spaces. For the other case, we prove that for finite-dimensional polyhedral Banach spaces, the approximate symmetry of the orthogonality is equivalent to some newly introduced…
Let C be a nonempty closed convex subset of a Banach space $E$. In this paper we introduce a new definition for variational inequality V I (C, B) on E that generalizes the analogue definition on Hilbert spaces. We generalize (u, v)-cocoercive mappings and v-strongly monotone mappings from Hilbert spaces to Banach spaces. Then we prove the generalized variational inequality V I (C, B) is singleton for (u, v)-cocoercive mappings under appropriate assumptions on Banach spaces that extends and improves [S. Saeidi, Comments on relaxed (u, v)-cocoercive mappings. Int. J. Nonlinear Anal. Appl. 1 (201…
Best approximation (BA) is an interesting field in functional analysis that has attracted a lot of attention from many researchers for a very long period of time up-to-date. Of greatest consideration is the characterization of the Chebyshev set (CS) which is a subset of a normed linear space (NLS) which contains unique BAs. However, a fundamental question remains unsolved to-date regarding the convexity of the CS in infinite NLS known as the CS problem. The question which has not been answered is: Is every CS in a NLS convex?. This question has not got any solution including the simplest form …
Rigidity of unit spheres in finite-dimensional real normed spaces
Department of Applied Mathematics, Graduate School of Science, Tokyo University of Science（东京理科大学理工学研究科应用数学系）；Katsushika Division, Institute of Arts and Sciences, Tokyo University of Science（东京理科大学理工学院葛饰校区）
