Examples of differentiable mappings into real or complex topological vector spaces with specific properties are given, which illustrate the differences between differential calculus in the locally convex and the non-locally convex case. In particular, for a suitable non-locally convex space E, we describe a smooth injection of R into E whose derivative vanishes identically; we present a complex C^\infty-map on the complex field C which is not given locally by its Taylor series, around any point; we present a complex C^1-map into a complete, non-locally convex topological vector space which is not C^2; and we present a compactly supported, non-zero, complex C^\infty-map from C to a suitable non-locally convex space.
In this paper, we establish several new inequalities for n- time differentiable mappings that are connected with the celebrated Hermite-Hadamard integral inequality.
In a recent paper [9], Ozdemir, Tunc and Akdemir defined two new classes of convex functions with which they proved some Hermite-Hadamard type inequalities. As an Open problem, they asked for conditions under which the composition of two functions belong to their newly defined class of convex functions and if Hermite-Hadamard type inequalities can be obtained. In this paper, we respond to the Open problems and prove some new Hermite-Hadamard inequalities for twice differentiable composition whose second derivative is $((h-s)_{2}, I)$-convex. Our results are applied to some special means of rea…
We show that if $X$ is a Banach space whose dual $X^{*}$ has an equivalent locally uniformly rotund (LUR) norm, then for every open convex $U\subseteq X$, for every $\varepsilon >0$, and for every continuous and convex function $f:U \rightarrow \mathbb{R}$ (not necessarily bounded on bounded sets) there exists a convex function $g:X \rightarrow \mathbb{R}$ of class $C^1(U)$ such that $f-\varepsilon\leq g\leq f$ on $U.$ We also show how the problem of global approximation of continuous (not necessarily bounded on bounded sets) and convex functions by $C^k$ smooth convex functions can be reduced…
Description of linear continuous functionals on a space of rapidly decreasing infinitely differentiable functions on an unbounded closed convex set in $\mathbb R^n$ in terms of their Fourier-Laplace transform is obtained.
We develop differential calculus of $C^{r,s}$-mappings on products of locally convex spaces and prove exponential laws for such mappings. As an application, we consider differential equations in Banach spaces depending on a parameter in a locally convex space. Under suitable assumptions, the associated flows are mappings of class $C^{r,s}$.
For each positive integer k, we describe a map f from the complex plane to a suitable non-complete complex locally convex space such that f is k times continuously complex differentiable but not k+1 times, and hence not complex analytic. We also describe a complex analytic map from l^1 to a suitable complete complex locally convex space which is unbounded on each non-empty open subset of l^1. Furthermore, we present a smooth map from the real line to a non-complete locally convex space which is not real analytic although it is given locally by its Taylor series around each point. As a byproduc…
Locally convex cones are generalization of locally convex spaces. The assertion, whether a barreled cone is an upper-barreled cone or not, was posed as a question in [A. Ranjbari, H. Saiflu, Projective and inductive limits in locally convex cones, J. Math. Anal. Appl. 332 (2) (2007) 1097-1108]. In this paper, we show that a barreled locally convex cone is not necessarily upper-barreled.
We show that a complemented subspace of a locally convex direct sum of an uncountable collection of Banach spaces is a locally convex direct sum of complemented subspaces of countable subsums. As a corollary we prove that a complemented subspace of a locally convex direct sum of arbitrary collection of $\ell_{1}(Γ)$-spaces is isomorphic to a locally convex direct sum of $\ell_{1}(Γ)$-spaces.
Abstract Equivalent conditions that make the convex subdifferential maximal monotone are investigated in the general settings of locally convex spaces.
To provide a solid analytic foundation for the module approach to conditional risk measures, our purpose is to establish a complete random convex analysis over random locally convex modules by simultaneously considering the two kinds of topologies (namely the $(\varepsilon,λ)$--topology and the locally $L^0$-- convex topology). This paper is focused on the part of separation and Fenchel-Moreau duality in random locally convex modules. The key point of this paper is to give the precise relation between random conjugate spaces of a random locally convex module under the two kinds of topologies, …
This paper constructs a counterexample showing that not every locally $L^0$--convex topology is necessarily induced by a family of $L^0$--seminorms. Random convex analysis is the analytic foundation for $L^0$--convex conditional risk measures, this counterexample, however, shows that a locally $L^0$--convex module is not a proper framework for random convex analysis. Further, this paper also gives a necessary and sufficient condition for a locally $L^0$--convex topology to be induced by a family of $L^0$--seminorms. Finally, we give some comments showing that based on random locally convex mod…
In this article we extend the notion of quasi-nilpotent equivalent operators, introduced by Colojoara and Foias \cite{co1} for Banach spaces, to the class of bounded operators on sequentially complete locally convex spaces.
Equivalent conditions that make the normal cone maximal monotone are investigated in the general settings of locally convex spaces. Some consequences such as Bishop Phelps and sum representability results are presented in the last part.
We develop a holomorphic functional calculus for (multivalued linear) operators on locally convex vector spaces. This includes the case of fractional powers along Lipschitz curves.
We construct a Lipschitz function on $\er^{2}$ which is locally convex on the complement of some totally disconnected compact set but not convex. Existence of such function disproves a theorem that appeared in a paper by L. Pasqualini and was also cited by other authors.
In this paper we study, in the relaxed context of locally convex spaces, intrinsic properties of monotone operators needed for the sum conjecture for maximal monotone operators to hold under classical interiority-type domain constraints.
We construct a complete locally convex topological vector space $X$ of countable algebraic dimension and a continuous linear operator $T:X\to X$ such that $T$ has no non-trivial closed invariant subspaces.
We treat some questions related to supercyclicity of continuous linear operators when acting in locally convex spaces. We extend results of Ansari and Bourdon and consider doubly power bounded operators in this general setting. Some examples are given.
In this brief note, we provide an example of non complete locally convex space $E$ with a $σ(E, E^*)$ closed bounded subset $C\subset E$, which is not $σ(E, E^*)$-compact, even if every $φ\in E^*$ attains its sup over $C$.
We prove a Desch-Schappacher type perturbation theorem for strongly continuous and locally equicontinuous one-parameter semigroups which are defined on a sequentially complete locally convex space.
In this paper, we continue to study random convex analysis. First, we introduce the notion of an $L^0$--pre--barreled module. Then, we develop the theory of random duality under the framework of a random locally convex module endowed with the locally $L^0$--convex topology in order to establish a characterization for a random locally convex module to be $L^0$--pre--barreled, in particular we prove that the model space $L^{p}_{\mathcal{F}}(\mathcal{E})$ employed in the module approach to conditional risk measures is $L^0$--pre--barreled, which forms the most difficult part of this paper. Finall…
This paper aims to study the dual of an extended locally convex space. In particular, we study the weak and weak* topologies as well as the topology of uniform convergence on bounded subsets of an extended locally convex space. As an application to function spaces, we show that the weak topology for the space C(X) of all real-valued continuous functions on a metric space (X,d) endowed with the topology of strong uniform convergence on bornology coincides with its finest locally convex topology if and only if the bornology is of all finite subsets of X.
We derive sufficient criteria for the uniqueness and existence of solutions of the abstract Cauchy problem in locally convex Hausdorff spaces. Our approach is based on a suitable notion of an asymptotic Laplace transform and extends results of Langenbruch beyond the class of Fréchet spaces.
The main purpose of this paper is to investigate $C$-distribution semigroups and $C$-ultradistribution semigroups in the setting of sequentially complete locally convex spaces. There are a few important theoretical novelties in this field and there are given some interesting examples.
The main subject in this paper are degenerate $C$-ultradistribution semigroups in barreled sequentially complete locally convex spaces. Here, the regularizing operator $C$ is not necessarily injective and the infinitesimal generator is multivalued linear operator. We also consider exponential degenerate $C$-ultradistribution semigroups.
As established by R T. Rockafellar, real valued convex-concave functions are generically differentiable. It this paper we shall show that for a convex-concave function defined on an open convex set $C \times D,$ there exist dense subsets ${\cal N}$ of $C$ and ${\cal M}$ of $D$ such that the partial derivative with respect to the first variable (resp. second variable) exists on ${\cal N} \times D$ (resp. $C \times {\cal M}$) and therefore the function is differentiable on ${\cal N} \times {\cal M}$. This is an interesting property of convex-concave functions and it does not hold for convex-conv…
The main purpose is to generalize a theorem of Arendt about uniqueness of $C_0$-semigroups from Banach space setting to the general locally convex vector spaces, more precisely, we show that cores are the only domains of uniqueness for $C_0$-semigroups on locally convex spaces. As an application, we find a necessary and sufficient condition for that the mass transport equation has one unique $L^1(\R^d,dx)$ weak solution.
Suppose that Q is a family of seminorms on a locally convex space E which determines the topology of E. We study the existence of Q-nonexpansive retractions for families of Q-nonexpansive mappings and prove that a separated and sequentially complete locally convex space $E$ that has the weak fixed point property, has the weak fixed point property for commuting separable semitopological semigroups of Q-nonexpansive mappings. This proves the Bruck's problem [5] for locally convex spaces. Moreover, we prove the existence of Q-nonexpansive retractions for the right amenable Q-nonexpansive semigrou…
We prove that for every closed locally convex subspace $E$ of $L_0$ and for any continuous linear operator $T$ from $L_0$ to $L_0/E$ there is a continuous linear operator $S$ from $L_0$ to $L_0$ such that $T = QS$ where $Q$ is the quotient map from $L_0$ to $L_0/E$.
In this paper we define the Radon-Nikodym class (RN class) of locally convex topological vector spaces. The RN class is characterized in terms of the Radon-Nikodym theorem for vector measures using integrable by seminorm derivatives. It is shown that the RNP class of all complete Hausdorff locally convex spaces possessing the Radon-Nikodym property is properly contained in the RN class. As an application we present a Radon-Nikodym theorem for multimeasures with respect to the Pettis integrable multifunctions in terms of the RN class.
The General Curve Lemma is a tool of Infinite-Dimensional Analysis, which enables refined studies of differentiability properties of mappings between real locally convex spaces. In this article, we generalize the General Curve Lemma in two ways: First, we remove the condition of local convexity in the real case. Second, we adapt the lemma to the case of curves in topological vector spaces over ultrametric fields.
Our main goal in this paper is to investigate the (q-)exponential $C$-distribution semigroups and (q-)exponential $C$-ultradistribution semigroups in the setting of sequentially complete locally convex spaces. We contribute to previous work and the work of many other authors, providing additionally plenty of various examples and applications of obtained results.
Locally $L^0$-convex modules were introduced in [D. Filipovic, M. Kupper, N. Vogelpoth. Separation and duality in locally $L^0$-convex modules. J. Funct. Anal. 256(12), 3996-4029 (2009)] as the analytic basis for the study of conditional risk measures. Later, the algebra of conditional sets was introduced in [S. Drapeau, A. Jamneshan, M. Karliczek, M. Kupper. The algebra of conditional sets and the concepts of conditional topology and compactness. J. Math. Anal. Appl. 437(1), 561-589 (2016)]. In this paper we study locally $L^0$-convex modules, and find exactly which subclass of locally $L^0$-…
Let $C(X,E)$ be the linear space of all continuous functions on a compact Hausdorff space $X$ with values in a locally convex space $E$. We characterize maps $T:C(X,E)\to C(Y,E)$ which satisfy $\mathrm{Ran}(TF-TG)\subset\mathrm{Ran}(F-G)$ for all $F,G\in C(X,E)$.
The Egoroff theorem for measurable $\bold X$-valued functions and operator-valued measures $\bold m: Σ\to L(\bold X, \bold Y)$, where $Σ$ is a $σ$-algebra of subsets of $T \neq \emptyset$ and $\bold X$, $\bold Y$ are both locally convex spaces, is proved. The measure is supposed to be atomic and the convergence of functions is net.
Suppose that $Q$ is a family of seminorms on a locally convex space $E$ which determines the topology of $E$. In this paper, first we define the notation of the $q$-duality mappings in locally convex spaces. Then we introduce an implicit method for finding an element of the set of fixed points of a $Q$-nonexpansive mapping. Then we prove the convergence of the proposed implicit scheme to a fixed point of the $Q$-nonexpansive mapping in $τ_{Q}$.
We establish some new common fixed point theorems of single-valued and multivalued mappings operating between complete ordered locally convex spaces under weaker assumptions. As an application, we prove a new minimax theorem of existence of a solution of a game.
We consider a space of infinitely smooth functions on an unbounded closed convex set in ${\mathbb R}^n$. It is shown that each function of this space can be extended to an entire function in ${\mathbb C}^n$ satisfying some prescribed growth condition. Description of linear continuous functionals on this space in terms of their Fourier-Laplace transform is obtained. Also a variant of the Paley-Wiener-Schwartz theorem for tempered distributions is given it the paper.
The purpose of this paper is to provide a characterization of the topological $L^0$-modules whose topology is induced by a family of $L^0$-seminorms using the gauge function for $L^0$-modules. Taking advantage of these ideas we will give a counterexample of a locally $L^0$-convex module whose topology is not induced by a family of $L^0$-seminorms.
The original M.Valdivia proof of his theorem on representation of the space $O_M$ uses results of S.Rolewicz concerning metric linear spaces and results of A.Grothendieck from his theory of topological tensor product. We present a more direct proof.
Locally $L^0$-convex modules were introduced in [D. Filipovic, M. Kupper, N. Vogelpoth. Separation and duality in locally $L^0$-convex modules. J. Funct. Anal. 256(12), 3996-4029 (2009)] as the analytic basis for the study of multi-period mathematical finance. Later, the algebra of conditional sets was introduced in [S. Drapeau, A. Jamneshan, M. Karliczek, M. Kupper. The algebra of conditional sets and the concepts of conditional topology and compactness. J. Math. Anal. Appl. 437(1), 561-589 (2016)]. By means of Boolean-valued models and its transfer principle we show that any known result on …
For the study of some typical problems in finance and economics, Žitković %[G. Žitković, Convex compactness and its applications, Math. Finan. Eco., 3(1)(2010) 1--12] introduced convex compactness and gave many remarkable applications. Recently, motivated by random convex optimization and random variational inequalities, Guo, et al introduced $L^0$-convex compactness, developed the related theory of $L^0$-convex compactness in random normed modules and further applied it to backward stochastic equations. %[T.X. Guo, et al, Two fixed point theorems in complete random normed modules and their ap…
A bounded linear operator $T$ on a Banach space $X$ is called a convex-cyclic operator if there exists a vector $x \in X$ such that the convex hull of $Orb(T, x)$ is dense in $X$. In this paper, for given an aperiodic element $g$ in a locally compact group $G$, we give some sufficient conditions for a weighted translation operator $T_{g,w}: f \mapsto w\cdot f*δ_g$ on $\mathfrak{L}^{p}(G)$ to be convex-cyclic. A necessary condition is also studied. At the end, to explain the obtained results, some examples are given.
The main purpose of this paper is to investigate degenerate $C$-distribution semigroups in the setting of barreled sequentially complete locally convex spaces. In our approach, the infinitesimal generator of a degenerate $C$-distribution semigroup is a multivalued linear operator and the regularizing operator $C$ is not necessarily injective. We provide a few important theoretical novelties, considering also exponential subclasses of degenerate $C$-distribution semigroups.
Let $L(X)$ be the free locally convex space over a Tychonoff space $X$. We show that the following assertions are equivalent: (i) $L(X)$ is $\ell_\infty$-barrelled, (ii) $L(X)$ is $\ell_\infty$-quasibarrelled, (iii) $L(X)$ is $c_0$-barrelled, (iv) $L(X)$ is $\aleph_0$-quasibarrelled, and (v) $X$ is a $P$-space. If $X$ is a non-discrete metrizable space, then $L(X)$ is $c_0$-quasibarrelled but it is neither $c_0$-barrelled nor $\ell_\infty$-quasibarrelled. We prove that $L(X)$ is a $(DF)$-space iff $X$ is a countable discrete space. We show that there is a countable Tychonoff space $X$ such tha…
Theoretically speaking, there are four kinds of possibilities to define the random conjugate space of a random locally convex module. The purpose of this paper is to prove that among the four kinds there are only two which are universally suitable for the current development of the theory of random conjugate spaces: in this process we also obtain a somewhat surprising and crucial result that for a random normed module with base $(Ω,{\cal F},P)$ such that $(Ω,{\cal F},P)$ is nonatomic then the random normed module is a totally disconnected topological space when it is endowed with the locally $…
In this paper we undertake a systematic study of those locally convex spaces $E$ such that $(E, w)$ is (linearly) Eberlein-Grothendieck, where $w$ is the weak topology of $E$. Let $C_{k}(X)$ be the space of continuous real-valued functions on a Tychonoff space $X$ endowed with the compact-open topology. The main results of our paper are: (1) For a first-countable space $X$ (in particular, for a metrizable $X$) the locally convex space $(C_{k}(X), w)$ is Eberlein-Grothendieck if and only if $X$ is both $σ$-compact and locally compact; (2) $(C_{k}(X), w)$ is linearly Eberlein-Grothendieck if and…
We prove a version of the Stokes formula for differential forms on locally convex spaces. The main tool used for proving this formula is the surface layer theorem proved in another paper by the author. Moreover, for differential forms of a Sobolev-type class relative to a differentiable measure, we compute the operator adjoint to the exterior differential in terms of standard operations of calculus of differential forms and the logarithmic derivative. Previously, this connection was established under essentially stronger assumptions.
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 …
Smooth functions $f:G\to E$ from a topological group $G$ to a locally convex space $E$ were considered by Riss (1953), Boseck, Czichowski and Rudolph (1981), Beltiţă and Nicolae (2015), and others, in varying degrees of generality. The space $C^\infty(G,E)$ of such functions carries a natural topology, the compact-open $C^\infty$-topology. For topological groups $G$ and $H$, we show that $C^\infty(G\times H,E)\cong C^\infty(G,C^\infty(H,E))$ as a locally convex space, whenever both $G$ and $H$ are metrizable or both $G$ and $H$ are locally compact. Likewise, $C^k(G, C^l(H,E))$ can be identifie…
In this paper, we introduce a new notion of representation for a locally convex partial *-algebraic module as a concrete space of maps. This is a continuation of our systematic study of locally convex partial *-algebraic modules, which are generalizations of inner product modules over C*-algebras.
