We introduce a new topological property called (*) and the corresponding class of topological spaces, which includes spaces with $G_δ$-diagonals and Gruenhage spaces. Using (*), we characterise those Banach spaces which admit equivalent strictly convex norms, and give an internal topological characterisation of those scattered compact spaces $K$, for which the dual Banach space $C(K)^*$ admits an equivalent strictly convex dual norm. We establish some relationships between (*) and other topological concepts, and the position of several well-known examples in this context. For instance, we show that $C(\mathcal{K})^*$ admits an equivalent strictly convex dual norm, where $\mathcal{K}$ is Kunen's compact space. Also, under the continuum hypothesis CH, we give an example of a compact scattered non-Gruenhage space having (*).
We prove that if K is a Gruenhage compact space then C(K)* admits an equivalent, strictly convex dual norm. As a corollary, we show that if X is a Banach space and X* is the |.|-closed linear span of K, where K is a Gruenhage compact in the w*-topology and |.| is equivalent to a coarser, w*-lower semicontinuous norm on X*, then X* admits an equivalent, strictly convex dual norm. We give a partial converse to the first result by showing that if T is a tree, then C(T)* admits an equivalent, strictly convex dual norm if and only if T is a Gruenhage space. Finally, we present some stability proper…
The Isbell, compact-open and point-open topologies on the set $C(X,\mathbb{R})$ of continuous real-valued maps can be represented as the dual topologies with respect to some collections $α(X)$ of compact families of open subsets of a topological space $X$. Those $α(X)$ for which addition is jointly continuous at the zero function in $C_α(X,\mathbb{R})$ are characterized, and sufficient conditions for translations to be continuous are found. As a result, collections $α(X)$ for which $C_α(X,\mathbb{R})$ is a topological vector space are defined canonically. The Isbell topology coincides with thi…
We study the class of compact convex subsets of a topological vector space which admits a strictly convex and lower semicontinuous function. We prove that such a compact set is embeddable in a strictly convex dual Banach space endowed with its weak$^*$ topology. In addition, we find exposed points where a strictly convex lower semicontinuous function is continuous.
It is shown that if a Banach space X has the weak Banach-Saks property and the weak fixed point property for nonexpansive mappings and Y satisfies property asymptotic (P) (which is weaker than the condition WCS(Y)>1), then the direct sum of X and Y endowed with a strictly monotone norm enjoys the weak fixed point property. The same conclusion is valid if X admits a 1-unconditional basis.
The aim of this paper is to give two complete and simple characterizations of Minkowski norms N on an arbitrary topological real vector space such that the sublevel sets of N are strictly convex. We first show that this property is equivalent to the continuity of N together with the fact that any open chord between two points of the boundary of the sublevel set N^{-1}([0, 1)) lies inside that set (geometric characterization). On the other hand, we prove that this is also the same as saying that N is continuous and that for an arbitrary real number $α$ > 1 the function N^$α$ is strictly convex …
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 $(X,weak)$…
In this paper, we continue the investigation of topological properties of unbounded norm (un-)topology in normed lattices. We characterize separability and second countability of un-topology in terms of properties of the underlying normed lattice. We apply our results to prove that an order continuous Banach function space $X$ over a semi-finite measure space is separable if and only if it has a $σ$-finite carrier and is separable with respect to the topology of local convergence in measure. We also address the question when a normed lattice is a normal space with respect to the un-topology.
Let X be a real normed vector space and dim X \ge 2. Let d>0 be a fixed real number. We prove that if x,y \in X and ||x-y||/d is a rational number then there exists a finite set {x,y} \subseteq S(x,y) \subseteq X with the following property: for each strictly convex Y of dimension 2 each map from S(x,y) to Y preserving the distance d preserves the distance between x and y. It implies that each map from X to Y that preserves the distance d is an isometry.
A Banach space (or its norm) is said to have the diameter $2$ property (D$2$P in short) if every nonempty relatively weakly open subset of its closed unit ball has diameter $2$. We construct an equivalent norm on $L_1[0,1]$ which is weakly midpoint locally uniformly rotund and has the D$2$P. We also prove that for Banach spaces admitting a norm-one finite-co-dimensional projection it is impossible to be uniformly rotund in every direction and at the same time have the D$2$P.
In this paper, we study $un$-dual (in symbol, $\ud{E}$) of Banach lattice $E$ and compare it with topological dual $E^*$. If $E^*$ has order continuous norm, then $E^* = \ud{E}$. We introduce and study weakly unbounded norm topology ($wun$-topology) on Banach lattices and compare it with weak topology and $uaw$-topology. In the final, we introduce and study $wun$-Dunford-Pettis opertors from a Banach lattice $E$ into Banach space $X$ and we investigate some of its properties and its relationships with others known operators.
We prove that in a certain class E of nonseparable Banach spaces the norm topology of the dual ball is definable in terms of its weak* topology. Thus, any weak* homeomorphism between duals balls of spaces in E is automatically norm-continuous.
In this paper, we study some topological characteristics of the n-normed spaces. We observe convergence sequences, closed sets, and bounded sets in the n-normed spaces using norms of quotient spaces that will be constructed. These norms will be a new viewpoint in observing the characteristics of the n-normed spaces. By using these norms, we also review the completeness of the n-normed spaces.
In this article we investigate the disjointly non-singular (DNS) operators. Following [8] we say that an operator $T$ from a Banach lattice $F$ into a Banach space $E$ is DNS, if no restriction of $T$ to a subspace generated by a disjoint sequence is strictly singular. We partially answer a question from [8] by showing that this class of operators forms an open subset of $L\left(F,E\right)$ as soon as $F$ is order continuous. Moreover, we show that in this case $T$ is DNS if and only if the norm topology is the minimal topology which is simultaneously stronger than the unbounded norm topology …
A net $(x_α)$ in a Banach lattice $X$ is said to un-converge to a vector $x$ if $\bigl\lVert\lvert x_α-x\rvert\wedge u\bigr\rVert\to 0$ for every $u\in X_+$. In this paper, we investigate un-topology, i.e., the topology that corresponds to un-convergence. We show that un-topology agrees with the norm topology iff $X$ has a strong unit. Un-topology is metrizable iff $X$ has a quasi-interior point. Suppose that $X$ is order continuous, then un-topology is locally convex iff $X$ is atomic. An order continuous Banach lattice $X$ is a KB-space iff its closed unit ball $B_X$ is un-complete. For a Ba…
We study some generalized metric properties of weak topologies when restricted to the unit sphere of some equivalent norm on a Banach space, and their relationships with other geometrical properties of norms. In case of dual Banach space $X^*$, we prove that there exists a dual norm such that its unit sphere is a Moore space for the weak$^*$-topology (has a G$_δ$-diagonal for the weak$^*$-topology, respectively) if, and only if, $X^*$ admits an equivalent weak$^*$-LUR dual norm (rotund dual norm, respectively).
Salas and Garcia introduced the concept of an extended locally convex space in [D. Salas and S. Tapia-Garcia. Extended seminorms and extended topological vector spaces. Topology and its Applications, 2016] which extends the idea of an extended normed space (introduced by Beer in G. Beer. Norms with infinite values. Journal of Convex Analysis, 2015). This article gives an attractive formulation of the finest locally convex topology of an extended locally convex space and provides a systematic study of the resulting locally convex space. As an application, we characterize the coincidence of the …
Trees are very agreeable objects to work with, offering a diversity of behaviour within a structure that is sufficiently simple to admit precise analysis. Thus we are able to offer fairly satisfactory necessary and sufficient conditions on a tree $Υ$ for the existence of equivalent LUR or strictly convex norms on $\C_0(Υ)$ and for norms with the Kadec Property. In particular, we show that for a {\sl finitely branching} tree $Υ$ the space $\C_0(Υ)$ admits a Kadec renorming. Since some finitely branching trees fail the condition for strictly convex renormability, we obtain an example of a Banach…
We learn mathematics subjectively and must apply it objectively. But sometimes, we apply it subjectively by using wrong intuitions which may be elusive to our eyes. The aim of this note is to disclose the secretes of two kinds of these false intuitions and the opportunities they may provide. We first discuss the wrong assumption which says that each topology is uniquely determined by studying a very bad phenomenon happening in dealing with weak* topology. Then, we consider the problem of completing a normed space under two comparable norms, one being smaller than the other. Here, we show that …
An algebra A of operators on a Banach space X is called strictly semi-transitive if for all non-zero x,y in X there exists an operator S in A such that Sx=y or Sy=x. We show that if A is norm-closed and strictly semi-transitive, then every A-invariant linear subspace is norm-closed. Moreover, Lat A is totally and well ordered by reverse inclusion. If X is complex and A is transitive and strictly semi-transitive, then A is WOT-dense in L(X). It is also shown that if A is an operator algebra on a complex Banach space with no invariant operator ranges, then A is WOT-dense in L(X). This generalize…
In this paper, we generalize the concept of unbounded norm (un) convergence: let $X$ be a normed lattice and $Y$ a vector lattice such that $X$ is an order dense ideal in $Y$; we say that a net $(y_α)$ un-converges to $y$ in $Y$ with respect to $X$ if $\Bigl\lVert\lvert y_α-y\rvert \wedge x\Bigr\rVert\to 0$ for every $x\in X_+$. We extend several known results about un-convergence and un-topology to this new setting. We consider the special case when $Y$ is the universal completion of $X$. If $Y=L_0(μ)$, the space of all $μ$-measurable functions, and $X$ is an order continuous Banach function …
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 prove that the topology induced by algebraic cone metric coincides with the topology induced by the metric obtained via a nonlinear scalarization function, i.e. any algebraic cone metric space is metrizable. Furthermore, the notion of algebraic cone normed space is introduced and also normability of the topology of this space is proved.
Using a technique of adjoining an order unit to a normed linear space, we have characterized strictly convex spaces among normed linear spaces and Hilbert spaces among strictly convex Banach spaces respectively. This leads to a generalization of spin factors and provides a new class of absolute order unit spaces.
Using the game approach to fragmentability, we give new and simpler proofs of the following known results: (a) If the Banach space admits an equivalent Kadec norm, then its weak topology is fragmented by a metric which is stronger than the norm topology. (b) If the Banach space admits an equivalent rotund norm, then its weak topology is fragmented by a metric. (c) If the Banach space is weakly locally uniformly rotund, then its weak topology is fragmented by a metric which is stronger than the norm topology.
We prove that an asymmetric normed space is never a Baire space if the topology induced by the asymmetric norm is not equivalent to the topology of a norm. More precisely, we show that a biBanach asymmetric normed space is a Baire space if and only if it is isomorphic to its associated normed space.
We study the class of Banach spaces $X$ such that the locally convex space $(X,μ(X,Y))$ is complete for every norming and norm-closed subspace $Y \subset X^*$, where $μ(X,Y)$ denotes the Mackey topology on $X$ associated to the dual pair $\langle X,Y\rangle$. Such Banach spaces are called fully Mackey complete. We show that fully Mackey completeness is implied by Efremov's property ($\mathcal{E}$) and, on the other hand, it prevents the existence of subspaces isomorphic to $\ell_1(ω_1)$. This extends previous results by Guirao, Montesinos and Zizler [J. Math. Anal. Appl. 445 (2017), 944-952] a…
An estimate for the norm of selfadjoint Toeplitz operators with a radial, bounded and integrable symbol is obtained. This emphasizes the fact that the norm of such operator is strictly less than the supremum norm of the symbol. Consequences for time-frequency localization operators are also given.
We prove that a Banach space of continuous functions $C(K)$ has a renorming that is uniformly rotund in every direction (URED) if and only if the compact space $K$ supports a strictly positive measure
The main result: the dual of separable Banach space $X$ contains a total subspace which is not norming over any infinite dimensional subspace of $X$ if and only if $X$ has a nonquasireflexive quotient space with the strictly singular quotient mapping.
We show that a Banach space with numerical index one cannot enjoy good convexity or smoothness properties unless it is one-dimensional. For instance, it has no WLUR points in its unit ball, its norm is not Frechet smooth and its dual norm is neither smooth nor strictly convex. Actually, these results also hold if the space has the (strictly weaker) alternative Daugavet property. We construct a (non-complete) strictly convex predual of an infinite-dimensional $L_1$ space (which satisfies a property called lushness which implies numerical index~1). On the other hand, we show that a lush real Ban…
For each $n \in \mathbb{N}$ a Banach space $\mathfrak{X}_{0,1}^n$ is constructed is having the property that every normalized weakly null sequence generates either a $c_0$ or $\ell_1$ spreading models and every infinite dimensional subspace has weakly null sequences generating both $c_0$ and $\ell_1$ spreading models. The space $\mathfrak{X}_{0,1}^n$ is also quasiminimal and for every infinite dimensional closed subspace $Y$ of $\mathfrak{X}_{0,1}^n$, for every $S_1,S_2,\ldots,S_{n+1}$ strictly singular operators on $Y$, the operator $S_1S_2\cdots S_{n+1}$ is compact. Moreover, for every subsp…
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 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 the recent paper \cite{Hos}, surjective isometries, not necessarily linear, $T: {\rm AC}(X,E) \longrightarrow {\rm AC}(Y,F)$ between vector-valued absolutely continuous functions on compact subsets $X$ and $Y$ of the real line, has been described. The target spaces $E$ and $F$ are strictly convex normed spaces. In this paper, we assume that $X$ and $Y$ are compact Hausdorff spaces and $E$ and $F$ are normed spaces, which are not assumed to be strictly convex. We describe (with a short proof) surjective isometries $T: (A,\|\cdot\|_A) \longrightarrow (B,\|\cdot\|_B)$ between certain normed su…
We present a new method for constructing $C_0$-semigroups for which properties of the resolvent of the generator and continuity properties of the semigroup in the operator-norm topology are controlled simultaneously. It allows us to show that a) there exists a $C_0$-semigroup which is continuous in the operator-norm topology for no $t \in [0,1]$ such that the resolvent of its generator has a logarithmic decay at infinity along vertical lines; b) there exists a $C_0$-semigroup which is continuous in the operator-norm topology for no $t \in \mathbb R_+$ such that the resolvent of its generator h…
We introduce the concepts of max-closedness and numeraires of convex subsets in the nonnegative orthant of the topological vector space of all random variables built over a probability space, equipped with a topology consistent with convergence in probability. Max-closedness asks that maximal elements of the closure of a set already lie on the set. We discuss how numeraires arise naturally as strictly positive optimisers of certain concave monotone maximisation problems. It is further shown that the set of numeraires of a convex, max-closed and bounded set of of nonnegative random variables th…
