Separable Banach Space Research Articles

A topological characterisation of Haar null convex sets

In R d \mathbb {R}^d , a closed, convex set has zero Lebesgue measure if and only its interior is empty. More generally, in separable, reflexive Banach spaces, closed and convex sets are Haar null if and only if their interior is empty. We extend this facts by showing that a closed, convex set in a separable Banach space is Haar null if and only if its weak ∗ ^* closure in the second dual has empty interior with respect to the norm topology. It then follows that, in the metric space of all nonempty, closed, convex and bounded subsets of a separable Banach space, converging sequences of Haar null sets have Haar null limits.

The general class of Wasserstein Sobolev spaces: density of cylinder functions, reflexivity, uniform convexity and Clarkson’s inequalities

We show that the algebra of cylinder functions in the Wasserstein Sobolev space H^{1,q}(mathcal {P}_p(X,textsf{d}), W_{p, textsf{d}}, mathfrak {m}) generated by a finite and positive Borel measure mathfrak {m} on the (p,textsf{d})-Wasserstein space (mathcal {P}_p(X, textsf{d}), W_{p, textsf{d}}) on a complete and separable metric space (X,textsf{d}) is dense in energy. As an application, we prove that, in case the underlying metric space is a separable Banach space mathbb {B}, then the Wasserstein Sobolev space is reflexive (resp. uniformly convex) if mathbb {B} is reflexive (resp. if the dual of mathbb {B} is uniformly convex). Finally, we also provide sufficient conditions for the validity of Clarkson’s type inequalities in the Wasserstein Sobolev space.

On Successive Approximations for Compact-Valued Nonexpansive Mappings

We show that for a given initial point the typical, in the sense of Baire category, nonexpansive compact valued mapping F has the following properties: there is a unique sequence of successive approximations and this sequence converges to a fixed point of F. In the case of separable Banach spaces we show that for the typical mapping there is a residual set of initial points that have a unique trajectory.

On the dominated convergence theorem for the Kurzweil–Stieltjes integral

AbstractThis paper is concerned with a version of the Lebesgue dominated convergence theorem (DCT) which has been stated for the Kurzweil–Stieltjes integral of real functions. Our objective in this work is to analyze the extension of this result to include vector functions with values in Banach spaces. We establish that the mentioned convergence theorem for the Kurzweil–Stieltjes integral can be formulated in weaker versions for reflexive and separable Banach spaces, and spaces having the Schur property, nonetheless it is not verified in the general case.

Banach–Stone theorems for algebras of holomorphic germs on spaces with the approximation property

AbstractIt is shown that if and are balanced compact determining subsets of separable Banach spaces with the approximation property, then the algebras of holomorphic germs and are topologically algebra isomorphic if, and only if, and (the polynomial convex hulls of K and L) are biholomorphically equivalent.

Compact Hölder retractions and nearest point maps

In this paper, an α-Hölder retraction from any separable Banach space onto a compact convex subset whose closed linear span is the whole space is constructed for every positive α<1. This constitutes a positive solution to a Hölder version of a question raised by Godefroy and Ozawa. In fact, compact convex sets are found to be absolute α-Hölder retracts under certain assumption of flatness.

The existence of UFO implies projectively universal morphisms

Let C \mathcal C be a concrete category. We prove that if C \mathcal {C} admits a universally free object F \mathsf F , then there is a projectively universal morphism u : F → F u\colon \mathsf F\to \mathsf F , i.e., a morphism u u such that for any B ∈ C B\in \mathcal {C} and τ ∈ Mor ⁡ ( B ) \tau \in \operatorname {Mor}(B) there exists an epimorphism π ∈ Mor ⁡ ( F , B ) \pi \in \operatorname {Mor}(\mathsf F, B) such that π τ = u π \pi \tau = u \pi . This builds upon and extends various ideas by Darji and Matheron [Proc. Amer. Math. Soc. 145 (2017), pp. 251–265] who proved such a result for the category of separable Banach spaces with contractive operators as well as certain classes of dynamical systems on compact metric spaces. Specialising from our abstract setting, we conclude that the result applies to various categories of Banach spaces/lattices/algebras, C*-algebras, etc.

On sets of discontinuities of functions continuous on all lines

Answering a question asked by K. C. Ciesielski and T. Glatzer in 2013, we construct a $C^1$-smooth function $f$ on $[0,1]$ and a closed set $M \subset {\rm graph} f$ nowhere dense in ${\rm graph} f$ such that there does not exist any linearly continuous function on ${\mathbb R}^2$ (i.e., function continuous on all lines) which is discontinuous at each point of $M$. We substantially use a recent full characterization of sets of discontinuity points of linearly continuous functions on ${\mathbb R}^n$ proved by T. Banakh and O. Maslyuchenko in 2020. As an easy consequence of our result, we prove that the necessary condition for such sets of discontinuities proved by S. G. Slobodnik in 1976 is not sufficient. We also prove an analogue of this Slobodnik's result in separable Banach spaces.

Renormings preserving local geometry at countably many points in spheres of Banach spaces and applications

We develop tools to produce equivalent norms with specific local geometry around infinitely many points in the sphere of a Banach space via an inductive procedure. We combine this process with smoothness results and techniques to solve two open problems posed in the recently published monograph [7] by A. J. Guirao, V. Montesinos and V. Zizler. Specifically, on the one hand we construct in every separable Banach space admitting a Ck-smooth norm an equivalent norm which is Ck-smooth but fails to be uniformly Gâteaux in any direction; and on the other hand we produce in c0(Γ) for any infinite Γ a C∞-smooth norm whose ball is dentable but whose sphere lacks any extreme points.

Random fixed point results for generalized asymptotically nonexpansive random operators

In this paper, we define an implicit random iterative process with errors for three finite families of generalized asymptotically nonexpansive random operators. We also prove some convergence theorems using this iteration method in separable Banach spaces.

Upper semicontinuous convex-valued selectors of a Nemytskii operator with nonconvex values and evolution inclusions with maximal monotone operators

On the space of continuous functions, we consider a multivalued superposition operator, a Nemytskii operator valued in the space of p-integrable functions with 1≤p<∞. The Nemytskii operator is generated by a multivalued mapping defined on the direct product of a segment of the real line and a separable reflexive Banach space and having closed values in this space. At each point of the phase variable, the mapping either has a closed graph and is convex-valued or it is lower semicontinuous in some neighborhood of this point. In recent years, such mappings are called mappings with mixed semicontinuity properties. We prove that the restriction of the Nemytskii operator onto any compact subset of the space of continuous functions with the sup-norm has a multivalued selector with convex closed values that is upper semicontinuous in the weak topology of the space of p-integrable functions. This result generalizes some well-known results of the author valid in the case when the multivalued mapping generating the Nemytskii operator has compact values. A multivalued selector theorem is used to study an evolution inclusion in a separable Hilbert space with a time-dependent maximal monotone operator and a composite perturbation that is the sum of two multivalued mappings. The values of the first multivalued mapping are closed, bounded, not necessarily convex sets. The second multivalued mapping has closed values and possesses mixed semicontinuity properties. The existence of an absolutely continuous solution is proved. The proof is based on the theorem on an upper semicontinuous convex-valued selector and the author's theorem on selectors continuous in parameter and passing through fixed points of a parameter-dependent multivalued mapping with closed, nonconvex, decomposable values in the space of p-integrable functions as well as on the classical Ky Fan's fixed point theorem. Our approach is fairly simple, concise and clear as compared to other known approaches based on discretization.

Approximate Controllability of Non-autonomous Second Order Impulsive Functional Evolution Equations in Banach Spaces

This article investigates the approximate controllability of second order non-autonomous functional evolution equations involving non-instantaneous impulses and nonlocal conditions. First, we discuss the approximate controllability of second order linear system in detail, which lacks in the existing literature. Then, we derive sufficient conditions for approximate controllability of our system in separable reflexive Banach spaces via linear evolution operator, resolvent operator conditions, and Schauder’s fixed point theorem. Moreover, in this paper, we define proper identification of resolvent operator in Banach spaces. Finally, we provide two concrete examples to validate our results.

A Hadamard fractional total variation-Gaussian (HFTG) prior for Bayesian inverse problems

This paper studies the infinite-dimensional Bayesian inference method with Hadamard fractional total variation-Gaussian (HFTG) prior for solving inverse problems. First, Hadamard fractional Sobolev space is established and proved to be a separable Banach space under some mild conditions. Afterwards, the HFTG prior is constructed in this separable fractional space, and the proposed novel hybrid prior not only captures the texture details of the region and avoids staircase effects, but also provides a complete theoretical analysis in the infinite dimensional Bayesian inversion. Based on the HFTG prior, the well-posedness and finite-dimensional approximation of the posterior measure of the Bayesian inverse problem are given, and samples are extracted from the posterior distribution using the standard preconditioned Crank-Nicolson (pCN) algorithm. Finally, numerical results under different models indicate that the Bayesian inference method with HFTG prior is effective and accurate.

Fredholm inversion around a singularity: Application to autoregressive time series in Banach space

&lt;abstract&gt;&lt;p&gt;This paper considers inverting a holomorphic Fredholm operator pencil. Specifically, we provide necessary and sufficient conditions for the inverse of a holomorphic Fredholm operator pencil to have a simple pole and a second order pole. Based on these results, a closed-form expression of the Laurent expansion of the inverse around an isolated singularity is obtained in each case. As an application, we also obtain a suitable extension of the Granger-Johansen representation theorem for random sequences taking values in a separable Banach space. Due to our closed-form expression of the inverse, we may fully characterize solutions to a given autoregressive law of motion except a term that depends on initial values.&lt;/p&gt;&lt;/abstract&gt;

Asymptotic smoothness and universality in Banach spaces

For $1 \lt p\leq \infty $, we study the complexity and the existence of universal spaces for two classes of separable Banach spaces, denoted $\mathsf{A}_p$ and $\mathsf{N}_p$, and related to asymptotic smoothness in Banach spaces. We show that each of the

Characterizations of ball-covering of separable Banach space and application

&lt;abstract&gt;&lt;p&gt;In this paper, we first prove that the space $ (X, \|\cdot\|) $ is separable if and only if for every $ \varepsilon\in $ $ (0, 1) $, there is a dense subset $ G $ of $ X^{*} $ and a $ w^{*} $-lower semicontinuous norm $ \|\cdot\|_{0} $ of $ X^{*} $ so that (1) the norm $ \|\cdot\|_{0} $ is Frechet differentiable at every point of $ G $ and $ d_{F}\|x^{*}\|_{0}\in X $ is a $ w^{*} $-strongly exposed point of $ B(X^{**}, \|\cdot\|_{0}) $ whenever $ x^{*}\in G $; (2) $ \left(1+\varepsilon^{2}\right) {\left\| {{x^{***}}} \right\|_0} \le \left\| {{x^{***}}} \right\| \le \left(1 + \varepsilon \right){\left\| {{x^{***}}} \right\|_0} $ for each $ x^{***}\in X^{***} $; (3) there exists $ \{x_{i}^{*}\}_{i = 1}^{\infty}\subset G $ such that ball-covering $ \{ B({x_{i}^{*}}, {r_i})\} _{i = 1}^\infty $ of $ (X^{*}, \|\cdot\|_{0}) $ is $ (1+\varepsilon)^{-1} $-off the origin and $ S(X^{*}, \|\cdot\|)\subset \cup_{i = 1}^{\infty}B({x_{i}^{*}}, {r_i}) $. Moreover, we also prove that if space $ X $ is weakly locally uniform convex, then the space $ X $ is separable if and only if $ X^{*} $ has the ball-covering property. As an application, we get that Orlicz sequence space $ l_{M} $ has the ball-covering property.&lt;/p&gt;&lt;/abstract&gt;

A viability result for Carath�odory non-convex differential inclusion in Banach spaces

This paper deals with the existence of solutions to the following differential inclusion: \(\dot{x}(t)\in F(t,x(t))\) a.e. on \([0, T[\) and \(x(t)\in K\), for all \(t \in [0, T]\), where \(F: [0, T]\times K \rightarrow 2^E\) is a Carath�odory multifunction and \(K\) is a closed subset of a separable Banach space \(E\).

Porosity phenomena of non-expansive, Banach space mappings

For any non-trivial convex and bounded subset C of a Banach space, we show that outside of a σ-porous subset of the space of non-expansive mappings C → C, all mappings have the maximal Lipschitz constant one witnessed locally at typical points of C. This extends a result of Bargetz and the author from separable Banach spaces to all Banach spaces and the proof given is completely independent. We further establish a fine relationship between the classes of exceptional sets involved in this statement, captured by the hierarchy of notions of ϕ-porosity.

On generic convergence of successive approximations of mappings with convex and compact point images

We study the generic behavior of the method of successive approximations for set-valued mappings in separable Banach spaces. We consider the case of nonexpansive mappings with convex and compact point images and show that for the typical such mapping and typical points of its domain the sequence of successive approximations is unique and converges to a fixed point of the mapping.

Continuous nowhere Hölder functions on ℤ_{𝕡}

K. Weierstrass (1872) was probably the first to present the existence of continuous nowhere differentiable functions (although B. Bolzano, in 1822, was the first to come up with such a construction). Almost a century later, V. Gurariĭ [Dokl. Akad. SSSR 167 (1966), pp. 971–973] observed that the family of continuous functions on [ 0 , 1 ] [0,1] that are differentiable at no point contains, except for the null function, an infinite dimensional vector space. Moreover, and among other recent contributions in this direction, S. Hencl [Proc. Amer. Math. Soc. 128 (2000), p. 3505–3511] generalized the previously mentioned result by proving the existence of isometrical embeddings of separable Banach spaces into the set of nowhere approximatively differentiable and nowhere Hölder functions. Here, we continue this ongoing research with the study of continuous nowhere Hölder functions, no longer defined in subsets of R \mathbb {R} , but in subsets of the p p -adic field Q p \mathbb {Q}_p .