An F-space Sampler
Many results for Banach spaces also hold for quasi-Banach spaces. One important such example is results depending on the Baire category theorem (BCT). We use the BCT to explore Lions problem for a quasi-Banach couple ( A 0 , A 1 ) . Lions problem, posed in 1960s, is to prove that different parameters ( θ , p ) produce different interpolation spaces ( A 0 , A 1 ) θ , p . We first establish conditions on A 0 and A 1 so that interpolation spaces of this couple are strictly intermediate spaces between A 0 + A 1 and A 0 ∩ A 1 . This result, together with a reiteration theorem, gives a partial solution to Lions problem for quasi-Banach couples. We then apply our interpolation result to (partially) answer a question posed by Pietsch. More precisely, we show that if p ≠ p ∗ the operator ideals L p , q ( a ) ( X , Y ) , L p ∗ , q ∗ ( a ) ( X , Y ) generated by approximation numbers are distinct. Moreover, for any fixed p , either all operator ideals L p , q ( a ) ( X , Y ) collapse into a unique space or they are pairwise distinct. We cite counterexamples which show that using interpolation spaces is not appropriate to solve Pietsch’s problem for operator ideals based on general s -numbers. However, the BCT can be used to prove a lethargy result for arbitrary s -numbers which guarantees that, under very minimal conditions on X , Y , the space L p , q ( s ) ( X , Y ) is strictly embedded into L A ( X , Y ) .
We present quasi‐Banach spaces which are closely related to the duals of combinatorial Banach spaces. More precisely, for a compact family F of finite subsets of ω we define a quasi‐norm ∥·∥F whose Banach envelope is the dual norm for the combinatorial space generated by F. Such quasi‐norms seem to be much easier to handle than the dual norms and yet the quasi‐Banach spaces induced by them share many properties with the dual spaces. We show that the quasi‐Banach spaces induced by large families (in the sense of Lopez‐Abad and Todorcevic) are ℓ1‐saturated and do not have the Schur property. In particular, this holds for the Schreier families.
For a convex polytope P P in R d \mathbb {R}^d containing 0 0 in its interior, it is proved an interpolative estimate for the Carleson operator generated by the partial sums of multiple Fourier series over all dilated sets of P P . This result is used to prove that this operator is bounded from the Arias-Reyna space Q A ( T d ) Q\!A(\mathbb {T}^d) to the weak L 1 L^{1} -space on the d d -dimensional torus T d \mathbb {T}^d . As a by-product, it follows that the sums over λ P \lambda P of the multiple Fourier series of every function in Q A ( T d ) Q\!A(\mathbb {T}^d) converge almost everywhere to the function as λ \lambda tends to + ∞ +\infty . It is shown that this statement holds for every function in a Lorentz space Λ φ ( T d ) \Lambda _\varphi (\mathbb {T}^d) , which contains the Orlicz space L ( log L ) d log log log L ( T d ) L(\log L)^{d} \log \log \log L(\mathbb {T}^d) and so L p ( T d ) L^p(\mathbb {T}^d) for all p ∈ ( 1 , ∞ ] p \in (1, \infty ] . In particular, this extends Fefferman’s result on the almost everywhere convergence of the of multiple Fourier series over polygon in R 2 \mathbb {R}^2 of functions in L p ( T 2 ) L^p(\mathbb {T}^2) .
We consider order isomorphic and order linearly isometric copies of in the quasi-normed Calderón–Lozanovskiĭ spaces . We present a number of theorems describing these copies in the natural language of suitable properties of the quasi-normed ideal space E and the non-decreasing Orlicz function . In particular, we will characterize quasi-normed Orlicz–Lorentz spaces having order isomorphic and order linearly isometric copies of . Our studies are conducted in a full possible generality due to the measure space and the Orlicz function.
We study a special type of infinite direct sums E ( X ) which can be seen as the amalgam spaces characterized by a local component given by a countable family X = X α α ∈ I of quasi-normed function spaces and by a global component E , which is a quasi-normed sequence space. We characterize some fundamental properties of E ( X ) such as completeness, Köthe-duality, order continuity and the Fatou property. We also provide its Banach function space characterization. Then, we apply our general results to the appropriate amalgamations of Lorentz (Orlicz) function spaces and Lebesgue sequence spaces. Moreover, for the Lorentz-type amalgams, we derive interpolation results and prove the boundedness of a class of sublinear integral operators whose kernels satisfy a size condition.
We consider the geometric structure of quasi‐normed Calderón–Lozanovskiĭ spaces. First, we study relations between the quasi‐norm and the quasi‐modular “near zero” and “near one,” which are fundamental for the theory. With their help, we provide a precise description of the basic monotonicity properties. In comparison with the well‐known normed case, we develop a number of new techniques and methods, among which the conditions Δε and Δ2−str play a crucial role. From our general results, we conclude the criteria for monotonicity properties in quasi‐normed Orlicz spaces, which are new even in this particular context. We consider both the function and the sequence case as well as we admit degenerated Orlicz functions, which provides us with a maximal class of spaces under consideration. We also discuss the applications of suitable properties to the best dominated approximation problems in quasi‐Banach lattices.
We introduce non-linear traces of the Choquet type and Sugeno type on a semifinite factor M as a non-commutative analog of the Choquet integral and Sugeno integral for non-additive measures. We need a weighted dimension function p↦α(τ(p)) for projections p∈M, which is an analog of a monotone measure. They have certain partial additivities. We show that these partial additivities characterize non-linear traces of both the Choquet type and Sugeno type, respectively. Based on the notion of generalized eigenvalues and singular values, we show that non-linear traces of the Choquet type are closely related to the Lorentz function spaces and the Lorentz operator spaces if the weight functions α are concave. For the algebras of compact operators and factors of type II, we completely determine the condition that the associated weighted Lp-spaces for the non-linear traces become quasi-normed spaces in terms of the weight functions α for any 0<p<∞. We also show that any non-linear trace of the Sugeno type yields a certain metric on the factor. This is an attempt at non-linear and non-commutative integration theory on semifinite factors.
We develop a set of techniques that enable us to effectively recover Besov rough analysis from p-variation rough analysis. Central to our approach are new metric groups, in which some objects in rough path theory that have been previously viewed as two-parameter can be considered as path increments. Furthermore, we develop highly precise Lipschitz estimates for Young and rough differential equations, both in the variation and Besov scale.
In this paper, we introduce the notion of a quasimodular and we prove that the respective Minkowski functional of the unit quasimodular ball becomes a quasinorm. In this way, we refer to and complete the well-known theory related to the notions of a modular and a convex modular that lead to the F-norm and to the norm, respectively. We use the obtained results to consider the basic properties of quasinormed Calderón–Lozanovskiĭ spaces Eφ, where the lower Matuszewska–Orlicz index αφ plays the key role. Our studies are conducted in a full possible generality.
We give a proof of the uniform boundedness principle for linear continuous maps from F -spaces into topological vector spaces which is elementary and also quite simple.
The purpose of this paper is to provide distributional estimates for the series of the form with being elements from noncommutative Lorentz spaces and being Rademacher functions. To this end, we introduce a novel class of operators that are closely related to the dual Cesáro operator and construct a new extrapolation theorem that is of independent interest.
The description of (commutative and noncommutative) ‐isometries has been studied thoroughly since the seminal work of Banach. In the present paper, we provide a complete description for the limiting case, isometries on noncommutative ‐spaces, which extends the Banach–Stone theorem and Kadison's theorem for isometries of von Neumann algebras. The result is new even in the commutative setting.
We characterize weak amenability of Banach algebras of the form and , where A is an arbitrary Banach algebra and . Specifically, we generalize the case of for a -algebra A. We also discuss amenability and contractibility of these algebras.
We study the boundedness of Hardy–Littlewood maximal function on the spaces defined in terms of Choquet integrals associated with weighted Bessel and Riesz capacities. As a consequence, we obtain a class of weighted Sobolev inequalities.
We discuss how countable subadditivity of operators can be derived from subadditivity under mild forms of continuity, and provide examples manifesting such circumstances.
The linear space of all continuous real-valued functions on a Tychonoff space X with the pointwise topology (induced from the product space R X ) is denoted by C p ( X ) . In this paper we continue the systematic study of sequences spaces c 0 and ℓ q (for 0 < q ≤ ∞ ) with the topology induced from R N (denoted by ( c 0 ) p and ( ℓ q ) p , respectively) and their role in the theory of C p ( X ) spaces. For every infinite Tychonoff space X we construct a subspace F of C p ( X ) that is isomorphic to ( c 0 ) p ; if X contains an infinite compact subset, then the copy F of ( c 0 ) p is closed in C p ( X ) . It follows that C p ( X ) contains a copy of ( ℓ q ) p for every 0 < q ≤ ∞ . We prove that for any infinite compact space X the space C p ( X ) contains no closed copy of ( ℓ q ) p for q ∈ ( 0 , 1 ] ∪ { ∞ } and no complemented copy for 0 < q ≤ ∞ . Relation with results of Talagrand, Haydon, Levy and Odell will be also discussed. Examples and open problems will be provided.
It is known that for 0 > p ≤ 1 0>p\leq 1 , every linear operator T : L p ( μ ) → L p ( λ ) T: L_p( \mu ) \to L_p(\lambda ) is norm-bounded if and only if it is regular. Recently, this has been generalized to operators from L 1 ( μ ) L_1(\mu ) into the Lebesgue-Bochner space L 1 ( λ ; X ) L_1(\lambda ;X) in the form of dominated operators: every linear operator T : L 1 ( μ ) → L 1 ( λ ; X ) T: L_1( \mu ) \to L_1(\lambda ;X) is norm-bounded if and only if it is dominated. Using another method of proof, we generalize this result to all indices 0 > p ≤ 1 0>p\leq 1 . Our result asserts that if X X is a p p -Banach space, then every linear operator T : L p ( μ ) → L p ( λ ; X ) T: L_p( \mu ) \to L_p(\lambda ;X) is norm-bounded if and only if there is a positive operator S : L 1 ( μ ) → L 1 ( λ ) S:L_1(\mu ) \to L_1(\lambda ) satisfying for every f ∈ L p ( μ ) f\in L_p(\mu ) , We also obtain as a consequence, a version of Grothendieck inequality for bounded linear operators from L p ( μ ) L_p(\mu ) into L p ( λ ; X ) L_p(\lambda ;X) for 0 > p ≤ 1 0>p\leq 1 .
In this paper, we deal with the reverse of the generalized triangle inequality of the second type in quasi-Banach spaces. More exactly, by using the concept of equivalent p-norms, we provide some necessary and sufficient conditions for n-tuples to satisfy the mentioned inequality. As applications, we improve some already known results and present some characterizations of p-Banach spaces among quasi-Banach spaces. In particular, we prove that a quasi-Banach space such as X is a p-Banach space if and only if for all (μ1,⋯,μn)∈Rn
satisfying μj>0
for some j and μi<0
for all i≠j
, the generalized triangle inequality of the second type ∑ni=1∥xi∥pμi≤∥∑ni=1xi∥p(xi∈X)
holds only with the assumption μj≥maxi∈{1,…,n}∖{j}{1,|μi|}
.
The goal of this paper is to develop new fixed points for quasi upper semicontinuous set-valued mappings and compact continuous (single-valued) mappings, and related applications for useful tools in nonlinear analysis by applying the best approximation approach for classes of semiclosed 1-set contractive set-valued mappings in locally p -convex and p -vector spaces for p ∈ ( 0 , 1 ] . In particular, we first develop general fixed point theorems for quasi upper semicontinuous set-valued and single-valued condensing mappings, which provide answers to the Schauder conjecture in the affirmative way under the setting of locally p -convex spaces and topological vector spaces for p ∈ ( 0 , 1 ] ; then the best approximation results for quasi upper semicontinuous and 1-set contractive set-valued mappings are established, which are used as tools to establish some new fixed points for nonself quasi upper semicontinuous set-valued mappings with either inward or outward set conditions under various boundary situations. The results established in this paper unify or improve corresponding results in the existing literature for nonlinear analysis, and they would be regarded as the continuation of the related work by Yuan (Fixed Point Theory Algorithms Sci. Eng. 2022:20, 2022)–(Fixed Point Theory Algorithms Sci. Eng. 2022:26, 2022) recently.
In this paper points of lower strict monotonicity, upper strict monotonicity, lower local uniform monotonicity and upper local uniform monotonicity of Orlicz function spaces equipped with a Mazur–Orlicz F-norm and generated by a monotone (non-convex in general) Orlicz function are studied. We show that both the necessary and sufficient conditions and the suitable theorems are formulated in full generality. Moreover, lower (upper) strict monotonicity and lower (upper) local uniform monotonicity theorems of the paper https://doi.org/10.1007/s00010-018-0615-y are revealed as corollaries in this paper.
For every cube we let be a quasi-Banach function space over Q such that , and for define
We study necessary and sufficient conditions on X such that
In particular, we give a full characterization of the embedding in terms of so-called sparse collections of cubes and we give easily checkable and rather weak sufficient conditions for the embedding . Our main theorems recover and improve all previously known results in this area.
The axioms of symmetry and indiscernibility of a metric function can be eliminated and still define a topological space with good enough properties. However, among all the axioms that constitute the notion of a metric, the subadditivity or triangular inequality is the one that has been more difficult to relax. Even so, and due to the potential applications in other fields such as Artificial Intelligence, several efforts have been made in this direction. The notion of quasi-Banach space or the notion of b-metric space, are examples of structures where the subadditivity has been replaced by a weaker inequality. Following this line, in this paper we consider the non-symmetric version of the so called strong b-metric spaces. We focus our attention on the possibility of extending semi-Lipschitz maps, and still preserve (a uniform multiple of ) the Lipschitz constant. As our main result, we prove that the possibility of obtaining such extensions characterizes the spaces satisfying the proposed weak versions of the triangular inequality. We also show how our results can be applied in Machine Learning to solve some technical issues, in particular, some problems related to the extension of Lipschitz functions with respect to the cosine distance.
Optimal transport has recently proved to be a useful tool in various machine learning applications, needing comparisons of probability measures. Among these, applications of distributionally robust optimization naturally involve Wasserstein distances in their models of uncertainty, capturing data shifts or worst-case scenarios. Inspired by the success of the regularization of Wasserstein distances in optimal transport, we study in this paper the regularization of Wassserstein distributionally robust optimization. First, we derive a general strong duality result of regularized Wasserstein distributionally robust problems. Second, we refine this duality result in the case of entropic regularization and provide an approximation result when the regularization parameters vanish.
We establish the mapping properties of the fractional integral operators on Morrey spaces built on rearrangement-invariant quasi-Banach function spaces. As applications of our main result, we have the mapping properties of the Riemann-Liouville fractional integral and the Weyl fractional integral on Morrey spaces built on rearrangement-invariant quasi-Banach function spaces. We also give an application of our main result on Orlicz-Morrey spaces.
The goal of this paper is to develop some new and useful tools for nonlinear analysis by applying the best approximation approach for classes of semiclosed 1-set contractive set-valued mappings in locally p-convex or p-vector spaces for p∈(0,1]. In particular, we first develop general fixed point theorems for both set-valued and single-valued condensing mappings which provide answers to the Schauder conjecture in the affirmative way under the setting of (locally p-convex) p-vector spaces, then the best approximation results for upper semi-continuous and 1-set contractive set-valued are established, which are used as tools to establish some new fixed points for non-self set-valued mappings with either inward or outward set conditions under various situations. These results unify or improve corresponding results in the existing literature for nonlinear analysis.
Many researchers in geometric functional analysis are unaware of algebraic aspects of the subject and the advances they have permitted in the last half century. This book, written by two world experts on homological methods in Banach space theory, gives functional analysts a new perspective on their field and new tools to tackle its problems. All techniques and constructions from homological algebra and category theory are introduced from scratch and illustrated with concrete examples at varying levels of sophistication. These techniques are then used to present both important classical results and powerful advances from recent years. Finally, the authors apply them to solve many old and new problems in the theory of (quasi-) Banach spaces and outline new lines of research. Containing a lot of material unavailable elsewhere in the literature, this book is the definitive resource for functional analysts who want to know what homological algebra can do for them.
We establish Arazy–Cwikel type properties for the family of couples , , and show that is a Calderón–Mityagin couple if and only if . Moreover, we identify interpolation orbits of elements with respect to this couple for all p and q such that and obtain a simple positive solution of a Levitina–Sukochev–Zanin problem, clarifying its connections with whether has the Calderón–Mityagin property or not.
The goal of this paper is to develop some fundamental and important nonlinear analysis for single-valued mappings under the framework of p -vector spaces, in particular, for locally p -convex spaces for 0 < p ≤ 1 . More precisely, based on the fixed point theorem of single-valued continuous condensing mappings in locally p -convex spaces as the starting point, we first establish best approximation results for (single-valued) continuous condensing mappings, which are then used to develop new results for three classes of nonlinear mappings consisting of 1) condensing; 2) 1-set contractive; and 3) semiclosed 1-set contractive mappings in locally p -convex spaces. Next they are used to establish the general principle for nonlinear alternative, Leray–Schauder alternative, fixed points for nonself mappings with different boundary conditions for nonlinear mappings from locally p -convex spaces, to nonexpansive mappings in uniformly convex Banach spaces, or locally convex spaces with the Opial condition. The results given by this paper not only include the corresponding ones in the existing literature as special cases, but are also expected to be useful tools for the development of new theory in nonlinear functional analysis and applications to the study of related nonlinear problems arising from practice under the general framework of p -vector spaces for 0 < p ≤ 1 .
Finally, the work presented by this paper focuses on the development of nonlinear analysis for single-valued (instead of set-valued) mappings for locally p -convex spaces. Essentially, it is indeed the continuation of the associated work given recently by Yuan (Fixed Point Theory Algorithms Sci. Eng. 2022:20, 2022); therein, the attention is given to the study of nonlinear analysis for set-valued mappings in locally p -convex spaces for 0 < p ≤ 1 .
The main aim of this paper is to develop a general approach, which allows to extend the basics of Brudnyi-Kruglyak interpolation theory to the realm of quasi-Banach lattices. We prove that all K-monotone quasi-Banach lattices with respect to a L-convex quasi-Banach lattice couple have in fact a stronger property of the so-called K(p, q)-monotonicity for some , which allows us to get their description by the real K-method. Moreover, we obtain a refined version of the K-divisibility property for Banach lattice couples and then prove an appropriate version of this property for L-convex quasi-Banach lattice couples. The results obtained are applied to refine interpolation properties of couples of sequence - and function -spaces, considered for the full range .
The goal of this paper is to establish some general topological results, Rothe’s principle and Leray–Schauder alternative for the fixed point equation in p-vector spaces which may not locally convex for . By the fact that when p=1, the p-vector spaces is the usual topological vector spaces, the new results established in this paper on topological results, Rothe type and Leray–Schauder alternative for fixed point equations unify or improve corresponding ones in the existing literature; and in particular, we expect that these new results would provide useful tools for the study of nonlinear problems under the framework of p-vector for nonlinear functional analysis.
The aim of this paper is to present new results obtained in the study of spaces with a Δ-symmetric Banach Algebra, we defined the δ-characters functional and discuss the space of measurable function(Lo(µ)) is Δ- symmetric and is Banach algebra. Also, we define sℂ(Lo(µ)) by depended on Lo(µ) and proved the spaces is satisfies new properties in symmetric Banach algebra and prove any function in sℂ(Lo(µ)) satisfies all conditions of state functional and found corresponding between the space sℂ(Lo(µ)) and the space of all o- characters functional δch(Lo(µ)) on Lo(µ) as well as well defined. Also, we proved the functional ψ from Æ into sℂ(Lo(µ)) is continuous and state functional if every non-zero element has an inverse. We proved the quotient sℂ(Lo(µ))/Ι is also symmetric Δ –Banach algebra.
It is known that the class of p -vector spaces ( 0 < p ≤ 1 ) is an important generalization of the usual norm spaces with rich topological and geometrical structure, but most tools and general principles with nature in nonlinearity have not been developed yet. The goal of this paper is to develop some useful tools in nonlinear analysis by applying the best approximation approach for the classes of 1-set contractive set-valued mappings in p -vector spaces. In particular, we first develop general fixed point theorems of compact (single-valued) continuous mappings for closed p -convex subsets, which also provide an answer to Schauder’s conjecture of 1930s in the affirmative way under the setting of topological vector spaces for 0 < p ≤ 1 . Then one best approximation result for upper semicontinuous and 1-set contractive set-valued mappings is established, which is used as a useful tool to establish fixed points of nonself set-valued mappings with either inward or outward set conditions and related various boundary conditions under the framework of locally p -convex spaces for 0 < p ≤ 1 . In addition, based on the framework for the study of nonlinear analysis obtained for set-valued mappings with closed p -convex values in this paper, we conclude that development of nonlinear analysis and related tools for singe-valued mappings in locally p -convex spaces for 0 < p ≤ 1 seems even more important, and can be done by the approach established in this paper.
We study properties of twisted unions of metric spaces introduced in [Johnson, Lindenstrauss, and Schechtman 1986], and in [Naor and Rabani 2017]. In particular, we prove that under certain natural mild assumptions twisted unions of L 1 -embeddable metric spaces also embed in L 1 with distortions bounded above by constants that do not depend on the metric spaces themselves, or on their size, but only on certain general parameters. This answers a question stated in [Naor 2015] and in [Naor and Rabani 2017].
In the second part of the paper we give new simple examples of metric spaces such that their every embedding into L p , 1 ≤ p < ∞, has distortion at least 3, but which are a union of two subsets, each isometrically embeddable in L p . This extends the result of [K. Makarychev and Y. Makarychev 2016] from Hilbert spaces to L p -spaces, 1 ≤ p < ∞.
We consider the stability of the orthogonal Jensen additive and quadratic equations in F-spaces, through applying and extending the approach to the proof of a 2010 result of W.Frchner and J.Sikorska, we presenting a new method to get the stability. Moreover, we work in a more general and natural condition than considered before by other antuors.
We prove that if the squares of two unconditional bases are equivalent up to a permutation, then the bases themselves are permutatively equivalent. This settles a twenty-five year-old question raised by Casazza and Kalton in [13]. Solving this problem provides a new paradigm to study the uniqueness of unconditional basis in the general framework of quasi-Banach spaces. Multiple examples are given to illustrate how to put in practice this theoretical scheme. Among the main applications of this principle we obtain the uniqueness of unconditional basis up to permutation of finite sums of spaces with this property, as well as the first addition to the scant list of the known Banach spaces with a unique unconditional bases up to permutation since [14].
This book is devoted to the study of stochastic measures (SMs). An SM is a sigma-additive in probability random function, defined on a sigma-algebra of sets. SMs can be generated by the increments of random processes from many important classes such as square-integrable martingales and fractional Brownian motion, as well as alpha-stable processes. SMs include many well-known stochastic integrators as partial cases.
General Stochastic Measures provides a comprehensive theoretical overview of SMs, including the basic properties of the integrals of real functions with respect to SMs. A number of results concerning the Besov regularity of SMs are presented, along with equations driven by SMs, types of solution approximation and the averaging principle. Integrals in the Hilbert space and symmetric integrals of random functions are also addressed.
The results from this book are applicable to a wide range of stochastic processes, making it a useful reference text for researchers and postgraduate or postdoctoral students who specialize in stochastic analysis.
Our goal in this paper is to advance the state of the art of the topic of uniqueness of unconditional basis. To that end we establish general conditions on a pair (X,Y) formed by a quasi-Banach space X and a Banach space Y which guarantee that every unconditional basis of their direct sum X⊕Y splits into unconditional bases of each summand. As application of our methods we obtain that, among others, the spaces Hp(Td)⊕T(2) and Hp(Td)⊕ℓ2, for p∈(0,1) and d∈N, have a unique unconditional basis (up to equivalence and permutation).
A topological abelian group G is said to have the quasi-convex compactness property (briefly, qcp) if the quasi-convex hull of every compact subset of G is again compact. In this paper we prove that there exist locally quasi-convex metrizable complete groups G which endowed with the weak topology associated to their character groups G∧, do not have the qcp. Thus, Krein’s Theorem, a well known result in the framework of locally convex spaces, cannot be fully extended to locally quasi-convex groups. Some features of the qcp are also studied.
We show that, under suitable conditions, an operator acting like a shift on some sequence space has a frequently hypercyclic random vector whose distribution is strongly mixing for the operator. This result will be applied to chaotic weighted shifts. We also apply it to every operator satisfying the Frequent Hypercyclicity Criterion, recovering a result of Murillo and Peris.
In this paper, the extension of isometries is considered for some operators between the unit spheres of two real p-normed spaces (0
We present a new approach to define a suitable integral for functions with values in quasi-Banach spaces. The integrals of Bochner and Riemann have deficiencies in the non-locally convex setting. The study of an integral for p -Banach spaces initiated by Vogt is neither totally satisfactory, since there are quasi-Banach spaces which are p -convex for all 0 < p < 1 , so it is not always possible to choose an optimal p to develop the integration. Our method puts the emphasis on the galb of the space, which permits a precise definition of its convexity. The integration works for all spaces of galbs known in the literature. We finish with a fundamental theorem of calculus for our integral.
This paper is devoted to providing a unifying approach to the study of the uniqueness of unconditional bases, up to equivalence and permutation, of infinite direct sums of quasi-Banach spaces. Our new approach to this type of problem permits to show that a wide class of vector-valued sequence spaces have a unique unconditional basis up to a permutation. In particular, solving a problem from Albiac and Leránoz (J Math Anal Appl 374(2):394–401, 2011. https://doi.org/10.1016/j.jmaa.2010.09.048) we show that if XX is quasi-Banach space with a strongly absolute unconditional basis then the infinite direct sum ℓ1(X) has a unique unconditional basis up to a permutation, even without knowing whether XX has a unique unconditional basis or not. Applications to the uniqueness of unconditional structure of infinite direct sums of non-locally convex Orlicz and Lorentz sequence spaces, among other classical spaces, are also obtained as a by-product of our work.
We collect several useful equivalent conditions for order continuity in F‐normed Köthe spaces. Next, we study uniform monotonicity of Orlicz function (sequence) spaces equipped with the Mazur–Orlicz F‐norm. We show necessary and sufficient conditions, separately. Moreover, we study the best dominated approximation problems in F‐normed Köthe spaces showing that order continuity and strict monotonicity are crucial tools in these problems. Finally, we apply the general results in Orlicz spaces. We focus mainly on these methods and techniques which are essentially different and more delicate in comparison to the case of normed Köthe spaces.
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