Subsets Of Lp Research Articles

The Szlenk index of 𝐿_{𝑝}(𝑋) and 𝐴_{𝑝}

Given a Banach space X X , a w ∗ w^* -compact subset of X ∗ X^* , and 1 > p > ∞ 1>p>\infty , we provide an optimal relationship between the Szlenk index of K K and the Szlenk index of an associated subset of L p ( X ) ∗ L_p(X)^* . As an application, given a Banach space X X , we prove an optimal estimate of the Szlenk index of L p ( X ) L_p(X) in terms of the Szlenk index of X X . This extends a result of Hájek and Schlumprecht to uncountable ordinals. More generally, given an operator A : X → Y A:X\to Y , we provide an estimate of the Szlenk index of the “pointwise A A ” operator A p : L p ( X ) → L p ( Y ) A_p:L_p(X)\to L_p(Y) in terms of the Szlenk index of A A .

New Modular Fixed-Point Theorem in the Variable Exponent Spaces ℓp(.)

In this work, we prove a fixed-point theorem in the variable exponent spaces ℓp(.), when p−=1 without further conditions. This result is new and adds more information regarding the modular structure of these spaces. To be more precise, our result concerns ρ-nonexpansive mappings defined on convex subsets of ℓp(.) that satisfy a specific condition which we call “condition of uniform decrease”.

Remark on norm compactness in [formula omitted

We prove a compactness criterion in Lp(μ,X): a subset of Lp(μ,X) is relatively norm compact iff the set of integrals of its functions over any measurable set is relatively norm compact, it satisfies the Fréchet oscillation restriction condition and it is p-uniformly integrable. The proof is elementary.

On embeddings of finite subsets of ℓ_{𝑝}

We study finite subsets of ℓ p \ell _p and show that, up to a nowhere dense and Haar null complement, all of them embed isometrically into any Banach space that uniformly contains ℓ p n \ell _p^n .

The geometry of two-valued subsets of Lp -spaces

Abstract Let 𝓜(Ω, μ) denote the algebra of all scalar-valued measurable functions on a measure space (Ω, μ). Let B ⊂ 𝓜(Ω, μ) be a set of finitely supported measurable functions such that the essential range of each f ∈ B is a subset of {0,1}. The main result of this paper shows that for any p ∈ (0, ∞), B has strict p-negative type when viewed as a metric subspace of L p (Ω, μ) if and only if B is an affinely independent subset of 𝓜(Ω, μ) (when 𝓜(Ω, μ) is considered as a real vector space). It follows that every two-valued (Schauder) basis of L p (Ω, μ) has strict p-negative type. For instance, for each p ∈ (0, ∞), the system of Walsh functions in L p [0,1] is seen to have strict p-negative type. The techniques developed in this paper also provide a systematic way to construct, for any p ∈ (2, ∞), subsets of L p (Ω, μ) that have p-negative type but not q-negative type for any q > p. Such sets preclude the existence of certain types of isometry into L p -spaces.

Lineability and spaceability for the weak form of Peano’s theorem and vector-valued sequence spaces

Two new applications of a technique for spaceability are given in this paper. For the first time this technique is used in the investigation of the algebraic genericity property of the weak form of Peano’s theorem on the existence of solutions of the ODE u ′ = f ( u ) u’=f(u) on c 0 c_0 . The space of all continuous vector fields f f on c 0 c_0 is proved to contain a closed c \mathfrak {c} -dimensional subspace formed by fields f f for which, except for the null field, the weak form of Peano’s theorem fails to be true. The second application generalizes known results on the existence of closed c \mathfrak {c} -dimensional subspaces inside certain subsets of ℓ p ( X ) \ell _p(X) -spaces, 0 > p > ∞ 0 > p > \infty , to the existence of closed subspaces of maximal dimension inside such subsets.

COMPACTNESS AND GENERALIZED APPROXIMATION SPACES

ABSTRACT We show that generalized approximation spaces can be used to describe the relatively compact sets of Banach spaces. This leads to compactness and convergence criteria in the approximation spaces themselves. If these spaces can be described with the help of moduli of smoothness, then the criteria can be formulated in terms of the moduli. As applications we give a generalization of Bernstein's theorem about existence of elements with prescribed best approximation errors, compactness criteria for operators, a criterion for compactness in Sobolev type spaces, and a generalization of Simon's compactness criterion for subsets of Lp -spaces of Banach-space-valued functions.

Orthogonal families of real sequences

For x, y ϵ ℝω define the inner productwhich may not be finite or even exist. We say that x and y are orthogonal if (x, y) converges and equals 0.Define lp to be the set of all x ϵ ℝω such thatFor Hilbert space, l2, any family of pairwise orthogonal sequences must be countable. For a good introduction to Hilbert space, see Retherford [4].Theorem 1. There exists a pairwise orthogonal family F of size continuum such that F is a subset of lp for every p > 2.It was already known that there exists a family of continuum many pairwise orthogonal elements of ℝω. A family F ⊆ ℝω∖0 of pairwise orthogonal sequences is orthogonally complete or a maximal orthogonal family iff the only element of ℝω orthogonal to every element of F is 0, the constant 0 sequence.It is somewhat surprising that Kunen's perfect set of orthogonal elements is maximal (a fact first asserted by Abian). MAD families, nonprincipal ultrafilters, and many other such maximal objects cannot be even Borel.Theorem 2. There exists a perfect maximal orthogonal family of elements of ℝω.Abian raised the question of what are the possible cardinalities of maximal orthogonal families.Theorem 3. In the Cohen real model there is a maximal orthogonal set in ℝω of cardinality ω1, but there is no maximal orthogonal set of cardinality κ with ω1 < κ < ϲ.By the Cohen real model we mean any model obtained by forcing with finite partial functions from γ to 2, where the ground model satisfies GCH and γω = γ.

Continuously translating vector-valued measures

Let G be a locally compact group and A an arbitrary Banach space. L p ( G , A ) {L^p}(G,A) will denote the space of p-integrable A-valued functions on G. M ( G , A ) M(G,A) will denote the space of regular A-valued Borel measures of bounded variation on G. In this paper, we characterise the relatively compact subsets of L p ( G , A ) {L^p}(G,A) . Using this result, we prove that if μ ∈ M ( G , A ) \mu \, \in \, M(G,A) , such that either x → μ x x\, \to \, {\mu _x} or x → x μ x{ \to _x}\mu is continuous, then μ ∈ L 1 ( G , A ) \mu \, \in \, {L^1}(G,A) .

Construction of non-linear semi-groups using product formulas

Under certain circumstances, the Trotter-Lie formulaWt=lim(Ut/nVt/n)n is used to construct a non-linear semi-groupWt on closed subsets ofLP, 1≦p<∞. In particular we consider the situation whereUt=etA is a positivity preservingC0 (linear) semi-group andVt is generated by a (non-linear) functionF with certain monotonicity properties. In general,A andF are “singular” onLp and no requirement is made that one of them be “relatively bounded” with respect to the other. The generator of the resulting semi-groupWt turns out to be an extension ofA +F restricted to a suitable domain.

Closed convex invariant subsets of 𝐿_{𝑝}(𝐺)

Let G be a locally compact group. We characterize in this paper closed convex subsets K of L p ( G ) , 1 ⩽ p &gt; ∞ {L_p}(G),1 \leqslant p &gt; \infty , that are invariant under all left or all right translations. We prove, among other things, that K = { 0 } K = \{ 0\} is the only nonempty compact (weakly compact) convex invariant subset of L p ( G ) ( L 1 ( G ) ) {L_p}(G)\;({L_1}(G)) . We also characterize affine continuous mappings from P 1 ( G ) {P_1}(G) into a bounded closed invariant subset of L p ( G ) {L_p}(G) which commute with translations, where P 1 ( G ) {P_1}(G) denotes the set of nonnegative functions in L 1 ( G ) {L_1}(G) of norm one. Our results have a number of applications to multipliers from L q ( G ) {L_q}(G) into L p ( G ) {L_p}(G) .

A unified approach to certain nonlinear initial value and boundary value problems

is an mth-order linear differential operator defined on some subset of LP(a, b) and Y is a lower-order nonlinear operator. This general theorem is then applied to three types of problems. In the first application we establish the existence of an absolutely continuous solution of the initial value problem,

The ε-entropy of some compact subsets of lp

The ε-entropy of some compact subsets of lp