Alaoglu Theorem Research Articles

Weakly compact sets in Orlicz–Bochner sequence spaces

AbstractIn this work, we give three kinds of criteria for weak sets in Orlicz–Bochner sequence spaces without constraints, conditions posited in each criterion are necessary and sufficient. As an application, we give criteria for weak sets in Orlicz sequence spaces. Well‐known conclusions are exhibited once more, such as Schur's theorem, Banach–Alaoglu's theorem, and the boundedly compact principle of finite dimension space. The results obtained show that the weak compactness may not be extrapolated straightforwardly from to , for example, .

The equivalency of the Banach-Alaoglu theorem and the axiom of choice

The Axiom of Choice can be used to prove the Banach-Alaoglu theorem, while a weakened form of the Axiom of Choice is required to prove the full strength of the Hahn-Banach theorem, which is equivalent to the Banach-Alaoglu theorem. Although the Banach-Alaoglu theorem and the full- strength Axiom of Choice are not exactly equivalent, they are closely related.
 The Banach-Alaoglu theorem is a compactness theorem whose proof mainly depends on Tychonoff's theorem. In this article, Banach-Alaoglu theorem is equivalent to the axiom of choice, has been proved.

Analysis and simulation of Korteweg-de Vries-Rosenau-regularised long-wave model via Galerkin finite element method

In this paper, a Galerkin finite element method is designed and analyzed to simulate the nonlinear Korteweg-de Vries-Rosenau-regularized long-wave (KdV-RRLW) model. We establish the existence and uniqueness results in H02(Ω) Sobolev space by applying the Banach–Alaoglu theorem. Using appropriate projection, we derive the error estimates of a semidiscrete scheme for the finite element solution of the KdV-RRLW model. Furthermore, a second-order Crank-Nicolson scheme is employed for the temporal discretization and obtain the optimal order of convergence in the maximum norm. Finally, several numerical examples are provided to visualize the nature of the wave phenomena in one and two dimensional spaces and demonstrate the robustness of the developed finite element algorithm.

English

We combine the techniques of sequence spaces and general Orlicz functions that are broader than the classical cases of N-functions. We give three criteria for the weakly compact sets in general Orlicz sequence spaces. One criterion is related to elements of dual spaces. Under the restriction of $$\mathop {\lim }\limits_{u \to 0} M(u)/u = 0$$ , we propose two other modular types that are convenient to use because they get rid of elements of dual spaces. Subsequently, by one of these two modular criteria, we see that a set A in Riesz spaces lp (1 < p < ∞) is relatively sequential weakly compact if and only if it is normed bounded, that says sup $$\mathop {{\rm{sup}}}\limits_{u \in A} \sum\limits_{i = 1}^\infty {{{\left| {u(i)} \right|}^p} < \infty } $$ . The result again confirms the conclusion of the Banach-Alaoglu theorem.

Nonstandard existence proofs for reaction diffusion equations

We give an existence proof for distribution solutions to a scalar reaction diffusion equation, with the aim of illustrating both the differences and the common ingredients of the nonstandard and standard approaches. In particular, our proof shows how the operation of taking the standard part of a nonstandard real number can replace several different compactness theorems, such as Ascoli's theorem and the Banach--Alaoglu theorem on weak$^*$-compactness of the unit ball in the dual of a Banach space.

Projection Theorems for the Rényi Divergence on $\alpha $ -Convex Sets

This paper studies forward and reverse projections for the Rényi divergence of order α E (0, ∞) on α-convex sets. The forward projection on such a set is motivated by some works of Tsallis et al. in statistical physics, and the reverse projection is motivated by robust statistics. In a recent work, van Erven and Harremoës proved a Pythagorean inequality for Rényi divergences on α-convex sets under the assumption that the forward projection exists. Continuing this study, a sufficient condition for the existence of a forward projection is proved for probability measures on a general alphabet. For α <b xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ϵ</b> (1, ∞), the proof relies on a new Apollonius theorem for the Hellinger divergence, and for α E (0, 1), the proof relies on the Banach- Alaoglu theorem from the functional analysis. Further projection results are then obtained in the finite alphabet setting. These include a projection theorem on a specific α-convex set, which is termed an α-linear family, generalizing a result by Csiszar to α ≠ 1. The solution to this problem yields a parametric family of probability measures, which turns out to be an extension of the exponential family, and it is termed an α-exponential family. An orthogonality relationship between the α-exponential and α-linear families is established, and it is used to turn the reverse projection on an α-exponential family into a forward projection on an α-linear family. This paper also proves a convergence result of an iterative procedure used to calculate the forward projection on an intersection of a finite number of α-linear families.

The algebra of conditional sets and the concepts of conditional topology and compactness

The concepts of a conditional set, a conditional inclusion relation and a conditional Cartesian product are introduced. The resulting conditional set theory is sufficiently rich in order to construct a conditional topology, a conditional real and functional analysis indicating the possibility of a mathematical discourse based on conditional sets. It is proved that the conditional power set is a complete Boolean algebra, and a conditional version of the axiom of choice, the ultrafilter lemma, Tychonoff's theorem, the Borel–Lebesgue theorem, the Hahn–Banach theorem, the Banach–Alaoglu theorem and the Krein–Šmulian theorem are shown.

On the defect of compactness in Banach spaces

Concentration compactness methods improve convergence for bounded sequences in Banach spaces beyond the weak-star convergence provided by the Banach–Alaoglu theorem. A further improvement of convergence, known as profile decomposition, is possible up to defect of compactness, a series of “elementary concentrations” defined relative to the action of some group of linear isometric operators. This note presents a general profile decomposition for uniformly convex and uniformly smooth Banach spaces, generalizing the result of one of the authors (S.S.) for Sobolev spaces and of the other (C.T. jointly with I. Schindler) for general Hilbert spaces. Unlike in the Hilbert space case, profile decomposition is based not on weak convergence, but on a different mode of convergence, called polar convergence, which coincides with weak convergence if and only if the norm satisfies the known Opial condition, used in the context of fixed point theory for nonexpansive maps.

Compactness in spaces of p-integrable functions with respect to a vector measure

We prove that, under some reasonable requirements, the unit balls of the spaces $L^p(m)$ and $L^\infty(m)$ of a vector measure of compact range $m$ are compact with respect to the topology $\tau_m$ of pointwise convergence of the integrals. This result can be considered as a generalization of the classical Alaoglu Theorem to spaces of $p$-integrable functions with respect to vector measures with relatively compact range. Some applications to the analysis of the Saks spaces defined by the norm topology and $\tau_m$ are given.

Weak topologies and compactness in asymmetric functional analysis

The classical Bourbaki–Alaoglu theorem asserts that the polar U∘ of a neighborhood of 0 in a locally convex topological vector space V is a weak⁎compact subset of the dual space V⁎. G. Plotkin has proved an analogous theorem in the framework of continuous d-cones, a kind of asymmetric version of topological vector spaces in Domain Theory.In this paper we extend Plotkin's results. We consider topological spaces X with a finitary continuous algebraic structure in the sense of universal algebra instead of a cone structure. Linear functionals are replaced by continuous algebra homomorphisms into a test algebra R replacing the reals. Our main result, Theorem 5.8, concerns the compactness of the space C⁎ of continuous homomorphisms from X to R under appropriate hypotheses. We exhibit conditions under which C⁎ is not only a space but also an algebra, as in the classical situation. This leads us to the notion of entropicity in the sense of universal algebra.The background for our investigation is Domain Theory and its use in denotational semantics (see [8]). Thus our spaces are strongly non-Hausdorff. This paper can be seen as a contribution to asymmetric topology and analysis.

Proving the Banach–Alaoglu Theorem via the Existence of the Stone–Čech Compactification

The Banach–Alaoglu theorem is an important result in functional analysis whose standard proof relies on Tychonoff's theorem. In this note, the theorem is proved by assuming the existence of the Stone–Čech compactification for completely regular topological spaces.

Accretive operators and Banach Alaoglu theorem in Linear 2-normed spaces

In this paper we introduce the concept of accretive operator in linear 2-normed spaces, focusing on the relationships and the various aspects of accretive, m-accretive and maximal accretive operators. We prove the analogous of Banach-Alaoglu theorem in linear 2- normed spaces, obtaining an equivalent definition for accretive operators in linear 2-normed spaces.

The relation of Banach-Alaoglu theorem and Banach-Bourbaki-Kakutani-Šmulian theorem in complete random normed modules to stratification structure

Let (Ω,A,μ) be a probability space, K the scalar field R of real numbers or C of complex numbers,and (S,X) a random normed space over K with base (ω,A,μ). Denote the support of (S,X) by E, namely E is the essential supremum of the set {A ∈ A: there exists an element p in S such that Xp(ω) > 0 for almost all ω in A}. In this paper, Banach-Alaoglu theorem in a random normed space is first established as follows: The random closed unit ball S*(1) = {f ∈ S*: X*f ⩽ 1} of the random conjugate space (S*,X*) of (S,X) is compact under the random weak star topology on (S*,X*) iff E∩A=: {E∩A | A ∈ A} is essentially purely μ-atomic (namely, there exists a disjoint family {An: n ∈ N} of at most countably many μ-atoms from E ∩ A such that E = ∪n=1∞An and for each element F in E ∩ A, there is an H in the σ-algebra generated by {An: n ∈ N} satisfying μ(FΔH) = 0), whose proof forces us to provide a key topological skill, and thus is much more involved than the corresponding classical case. Further, Banach-Bourbaki-Kakutani-Smulian (briefly, BBKS) theorem in a complete random normed module is established as follows: If (S,X) is a complete random normed module, then the random closed unit ball S(1) = {p ∈ S: Xp ⩽ 1} of (S,X) is compact under the random weak topology on (S,X) iff both (S,X) is random reflexive and E ∩ A is essentially purely μ-atomic. Our recent work shows that the famous classical James theorem still holds for an arbitrary complete random normed module, namely a complete random normed module is random reflexive iff the random norm of an arbitrary almost surely bounded random linear functional on it is attainable on its random closed unit ball, but this paper shows that the classical Banach-Alaoglu theorem and BBKS theorem do not hold universally for complete random normed modules unless they possess extremely simple stratification structure, namely their supports are essentially purely μ-atomic. Combining the James theorem and BBKS theorem in complete random normed modules leads directly to an interesting phenomenum: there exist many famous classical propositions that are mutually equivalent in the case of Banach spaces, some of which remain to be mutually equivalent in the context of arbitrary complete random normed modules, whereas the other of which are no longer equivalent to another in the context of arbitrary complete random normed modules unless the random normed modules in question possess extremely simple stratification structure. Such a phenomenum is, for the first time, discovered in the course of the development of random metric theory.

Borel complexity and computability of the Hahn–Banach Theorem

The classical Hahn–Banach Theorem states that any linear bounded functional defined on a linear subspace of a normed space admits a norm-preserving linear bounded extension to the whole space. The constructive and computational content of this theorem has been studied by Bishop, Bridges, Metakides, Nerode, Shore, Kalantari Downey, Ishihara and others and it is known that the theorem does not admit a general computable version. We prove a new computable version of this theorem without unrolling the classical proof of the theorem itself. More precisely, we study computability properties of the uniform extension operator which maps each functional and subspace to the set of corresponding extensions. It turns out that this operator is upper semi-computable in a well-defined sense. By applying a computable version of the Banach–Alaoglu Theorem we can show that computing a Hahn–Banach extension cannot be harder than finding a zero in a compact metric space. This allows us to conclude that the Hahn–Banach extension operator is \({\bf {\Sigma^{0}_{2}}}\) -computable while it is easy to see that it is not lower semi-computable in general. Moreover, we can derive computable versions of the Hahn–Banach Theorem for those functionals and subspaces which admit unique extensions.

A domain-theoretic Banach–Alaoglu theorem

We give a domain-theoretic analogue of the classical Banach–Alaoglu theorem, showing that the patch topology on the weak$*$ topology is compact. Various theorems follow concerning the stable compactness of spaces of valuations on a topological space. We conclude with reformulations of the patch topology in terms of polar sets or Minkowski functionals, showing, in particular, that the ‘sandwich set’ of linear functionals is compact.

An observation on realcompact spaces

We give a characterization of realcompact spaces in terms of nets. By using the technique of this characterization we give easy proofs of the Tychonoff Theorem and the Alaoglu Theorem.

On the Borel Complexity of Hahn-Banach Extensions

The classical Hahn-Banach Theorem states that any linear bounded functional defined on a linear subspace of a normed space admits a norm-preserving linear bounded extension to the whole space. The constructive and computational content of this theorem has been studied by Bishop, Bridges, Metakides, Nerode, Shore, Kalantari, Downey, Ishihara and others and it is known that the theorem does not admit a general computable version. We prove a new computable version of this theorem without unrolling the classical proof of the theorem itself. More precisely, we study computability properties of the uniform extension operator which maps each functional and subspace to the set of corresponding extensions. It turns out that this operator is upper semi-computable in a well-defined sense. By applying a computable version of the Banach-Alaoglu Theorem we can show that computing a Hahn-Banach extension cannot be harder than finding a zero on a compact metric space. This allows us to conclude that the Hahn-Banach extension operator is ∑20–computable while it is easy to see that it is not lower semi-computable in general. Moreover, we can derive computable versions of the Hahn-Banach Theorem for those functionals and subspaces which admit unique extensions.

Fuzzy Volterra Integral Equations: A Stacking Theorem Approach

New existence results are established for fuzzy Volterra equations with a “local” integrably boundedness condition using: (i) the Banach Alaoglu theorem; and (ii) the stacking theorem.

The Dual Space of an Asymmetric Normed Linear Space

Given an asymmetric normed linear space (X, q), we construct and study its dual space (X ∗ , q ∗). In particular, we show that (X ∗ , q ∗) is a biBanach semilinear space and prove that (X, q) can be identified as a subspace of its bidual by an isometric isomorphism. We also introduce and characterize the so-called weak* topology which is generated in a natural way by the relation between (X, q) and its dual, and an extension of the celebrated Alaoglu's theorem is obtained. Some parts of our theory are presented in the more general setting of the space LC (X, Y) of all linear continuous mappings from the asymmetric normed linear space X to the asymmetric normed linear space Y. In particular, we show that LC (X, Y) can be endowed with the structure of an asymmetric normed semilinear space and prove that it is a biBanach space if Y is so.

On weak compactness of bounded sets in Banach and locally convex spaces

We investigate the compactness of one class of bounded subsets in Banach and locally convex spaces. We obtain a generalization of the Banach-Alaoglu theorem to a class of subsets that are not polars of convex balanced neighborhoods of zero.