DOCSLIB.ORG

Appendix A. Banach Spaces Br(X) = {Y E X Illy

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Appendix A. Banach Spaces Br(X) = {Y E X Illy Appendix A. Banach Spaces In the appendices A, B and C we summarise the fundamental concepts and results we require from the theory of Banach spaces. For the proofs, we shall refer to well-known textbooks, and we give a limited selection at the end. We assume that the reader is familiar with the fundamental topological concepts (topology, neighborhood, basis of neighborhoods, open, contact point, etc.) and give full proofs only for a few results from the theory of Banach spaces which are of central significance for our applications. As a rule, we base our considerations on a real vector space. In order to be able to deal with topological concepts such as open sets, limit points, convergent sequences, Cauchy sequences, etc. in a vector space X, such as in the Euclidean space IRn , we will assume that a generalised "concept of length" or a "seminorm" p is given in X. A s eminorm is a function p : X -+ [0, 00 ) with the properties p(>.x) = 1>'lp(x) and p(x + y) :s:; p(x) + p(y) for all >. E IR and all x, y EX. If p( x) = 0 always implies x = 0, a seminorm is called a norm, which is written as p(x) = Ilxli. A vector space X together with a norm 11·11 defined for its elements is called a normed space. In a normed space (X, 11·11, we obtain a topology, and hence all topological concepts, by defining the following sets as open sets. A subset A c X is called open if, for every x E A, there exists an r = r x > 0 such that the open ball Br(x) = {y E X Illy - xII < r} with centre x and radius r belongs to A : Br(x) C A. In a normed space (X, II . II) a sequence (Xn)nElN converges to a point x E X if, and only if, for every t: > 0 there exists an n., such that IIx n - xII < t: holds for all n ~ n.,. A sequence (Xn)nElN is called a Cauchy sequence if, and only if, for every t: > 0 there exists an n., such that for all n, m ~ ne , IIx n -xmll < t: holds. A normed space in which every Cauchy sequence converges is said to be complete, or a Banach space. A Banach space is said to be separable if there exists a countable dense subset of it. Two norms 11·111 and 11·112 on a vector space X are said to be equivalent if there exist positive numbers c, c' such that IIxlh :s:; cllxll2 and IIxl12 :s:; c'lIxlll hold for all x EX. Equivalent norms generate the same topology. A subset M of a normed space (X, 11·11) is said to be bounded if the norm is bounded on M: 364 Appendix A. Banach Spaces sup{lIxlll x EM} < 00. Equivalent norms have identical bounded sets. Com­ plete normed spaces, which can be regarded as (infinite-dimensional) analo­ gous of the Euclidean spaces lRn, ~an be obtained in the following manner: let there be a scalar product (.,.) defined on a vector space X. This is a function (. , .) : X x X -+ lR with the following properties: (x, AlYl + A2Y2) = Al (x, Yl) + A2(x, Y2), (x, y) = (y, x) for all x, Yl, Y2, Y E X and all Aj E lR, (x, x) 2: 0, and (x, x) = 0 for all x EX only when x = O. A vector space X with a scalar product (. ,.) is called a pre-Hilbert space. In a pre-Hilbert space (X, (.,.)) one has a norm defined by x -+ IIxll = +~, called the norm induced by the scalar product. If the normed space (X, 11·11 = ..;r.1) is complete, then (X, (., .)) is called a Hilbert space. The Theorem of Jordan and von Neumann characterises those norms which are induced by a scalar product. The characteristic of these norms is the parallelogram law for all Xj EX. The Cauchy-Schwarz inequality l(x,y)1 :5l1xll'lIylI for all x,y E X holds in every pre-Hilbert space. The Projection Theorem is valid in every Hilbert space just as in any finite­ dimensional Euclidean space: let M be a subspace (which means a closed vector subspace) of a Hilbert space 1i. Then every vector x E 1i can be uniquely represented as x = U x + Vx where U x E M and Vx E M..L = {x E 1i I (x, y) = 0, Vy EM}, and one has I\vxl\ = c(x,M) = inf I\x - ull = Ilx - uxll· uEM We then write 1i = M ffi M..L and say that 1i is the direct orthogonal sum of M and M..L. If the orthogonal subspace M..L of M is finite dimensional, then we say that M has finite codimension. We now come to a few examples. Example A.I. The Euclidean Spaces lRn , n E IN. One can introduce the following norms in the vector space lRn : for an arbitrary number p E [1,00) and x E lR n one sets IIxli oo = sup IXil. l~i~n It follows from the inequalities IIxll oo :5 I\xl\p :5 nl/Pllxl\oo, x E lRn, that all these norms are equivalent. (lRn, 1\. lip) is a Banach space for 1 :5 p ~ +00. It is only for p = 2 that (lRn, 1\·lIp) is a Hilbert space with the scalar product Appendix A. Banach Spaces 365 n (x,y) = LXiYi. i=l Example A.2. The Sequence Spaces [P(JR), I ::s; p ::s; 00. Let lP(JR) denote the vector space of all sequences x = (Xn)nEIN of real numbers Xn for which I ::s; p < 00 and sup IXnl < 00 for p = +00 nEIN hold. Together with the norms II x li p = (~ IxnlP) liP, l::S;p<oo, Ilxll oo = sup Ixnl , nEIN respectively, the sequences spaces (lP(JR), II· lip) are complete normed spaces. In this case different values of p give rise to nonequivalent norms. Once again, only the sequence space z2(JR) is a Hilbert space. Its scalar product is (x, y) = 2::::='=1 xnYn' For I ::s; P ::s; +00 the dual exponent p' corresponding to p is defined by the equation lip' + lip = 1. For all x E IP(JR) and all Y E [P' (JR), one has x . Y = (XnYn)nElN E 11(JR) and 00 Ilx . ylh = L IXnYnl ::s; IlxllpllYllp' (Holder's Inequality). n=l Example A.3. Spaces of Continuous Functions. Let K denote a compact topo­ logical space (see also Appendix C) and C(K; JR) = C(K), be the set of all continuous functions f : K -t JR. One knows that C(K) is a real vector space in a natural way and IlfllK = sUPxEK If(x)1 defines a norm on C(K). C(K) is complete with respect to this norm, and is thus a Banach space but not a Hilbert space. Example A.4. Spaces of Differentiable Functions. For a compact interval f = [a, b], let Cl(f; IRn) be the space of all n-tuples of real functions continuously differentiable in (a, b), which together with their derivatives possess continuous extensions to the boundary points t = a and t = b. The real vector space Cl(f, IRn) is a Banach space with respect to the norm Ilqlh,l = sup{lq(t)l, 14(t)l} . tEl For q(t) = (ql (t), ... , qn(t)), qi E Cl(f, IR), let q(t) = (ql(t), ... ,qn(t)), qi(t) = ~i(t) 366 Appendix A. Banach Spaces and Iq(t)1 = (~lqi(t)12) 1/2 A norm equivalent to II . Ih.r is given by the following: Example A.5. The Lebesgue Spaces LP(lRn). For every real number p 2 1 let CP (lR n) = {J : lRn ---+ lR I I measurable, II I summable}. £P (lRn) is a real vector space and I ---+ 1I/11p where II/l1p = (in I/(x)IP dX) lip is a seminorm on CP(lRn). One has II/l1p = 0 {:} I = 0 almost everywhere. Denoting by N the subspace of all functions from CP(lRn) which vanish almost everywhere, the quotient space together with the norm induced by 1I·llp (which is again denoted by II· lip) is a Banach space (Theorem 01 Riesz-Fischer). Similarly one defines for p = +00, COO (lRn) = {J : lRn ---+ lR I I measurable and bounded almost everywhere} 11/1100 = inf{c Ie> 0, and apart from a null set I/(x)1 ~ c holds} and, further, Then Loo(lRn) is also a Banach space. Just as in the case of sequence spaces, Holder's inequality holds here also, which means that if I E LP(lRn), 9 E U ' (lRn) with p' being the dual exponent to p 21, then I· 9 E Ll(lRn), and one has Once again we obtain a Hilbert space only for p = 2. L2(lRn) is a Hilbert space with the scalar product (f,gh = { I(x)g(x)dx. lrn. n The norm thereby induced is 11·112. (In our notation we do not make explicit the distinction between the equivalence classes and the representatives from the equivalence classes.) Appendix A. Banach Spaces 367 Replacing the space IR n by a measurable set G ~ IR n one obtains the Banach spaces LP(G), 1 ::; p ::; +00, of Lebesgue. The Banach spaces LP(G) with 1 ::; p < 00 are separable. For real vector spaces Xl and X 2 , let L(XI' X 2 ) be the collection of all linear maps or operators from X I to X 2.
Load more

Recommended publications
  • Duality for Outer $ L^ P \Mu (\Ell^ R) $ Spaces and Relation to Tent Spaces
    p r DUALITY FOR OUTER Lµpℓ q SPACES AND RELATION TO TENT SPACES MARCO FRACCAROLI p r Abstract. We prove that the outer Lµpℓ q spaces, introduced by Do and Thiele, are isomorphic to Banach spaces, and we show the expected duality properties between them for 1 ă p ď8, 1 ď r ă8 or p “ r P t1, 8u uniformly in the finite setting. In the case p “ 1, 1 ă r ď8, we exhibit a counterexample to uniformity. We show that in the upper half space setting these properties hold true in the full range 1 ď p,r ď8. These results are obtained via greedy p r decompositions of functions in Lµpℓ q. As a consequence, we establish the p p r equivalence between the classical tent spaces Tr and the outer Lµpℓ q spaces in the upper half space. Finally, we give a full classification of weak and strong type estimates for a class of embedding maps to the upper half space with a fractional scale factor for functions on Rd. 1. Introduction The Lp theory for outer measure spaces discussed in [13] generalizes the classical product, or iteration, of weighted Lp quasi-norms. Since we are mainly interested in positive objects, we assume every function to be nonnegative unless explicitly stated. We first focus on the finite setting. On the Cartesian product X of two finite sets equipped with strictly positive weights pY,µq, pZ,νq, we can define the classical product, or iterated, L8Lr,LpLr spaces for 0 ă p, r ă8 by the quasi-norms 1 r r kfkL8ppY,µq,LrpZ,νqq “ supp νpzqfpy,zq q yPY zÿPZ ´1 r 1 “ suppµpyq ωpy,zqfpy,zq q r , yPY zÿPZ p 1 r r p kfkLpppY,µq,LrpZ,νqq “ p µpyqp νpzqfpy,zq q q , arXiv:2001.05903v1 [math.CA] 16 Jan 2020 yÿPY zÿPZ where we denote by ω “ µ b ν the induced weight on X.
    [Show full text]
  • ON SEQUENCE SPACES DEFINED by the DOMAIN of TRIBONACCI MATRIX in C0 and C Taja Yaying and Merve ˙Ilkhan Kara 1. Introduction Th
    Korean J. Math. 29 (2021), No. 1, pp. 25{40 http://dx.doi.org/10.11568/kjm.2021.29.1.25 ON SEQUENCE SPACES DEFINED BY THE DOMAIN OF TRIBONACCI MATRIX IN c0 AND c Taja Yaying∗;y and Merve Ilkhan_ Kara Abstract. In this article we introduce tribonacci sequence spaces c0(T ) and c(T ) derived by the domain of a newly defined regular tribonacci matrix T: We give some topological properties, inclusion relations, obtain the Schauder basis and determine α−; β− and γ− duals of the spaces c0(T ) and c(T ): We characterize certain matrix classes (c0(T );Y ) and (c(T );Y ); where Y is any of the spaces c0; c or `1: Finally, using Hausdorff measure of non-compactness we characterize certain class of compact operators on the space c0(T ): 1. Introduction Throughout the paper N = f0; 1; 2; 3;:::g and w is the space of all real valued sequences. By `1; c0 and c; we mean the spaces all bounded, null and convergent sequences, respectively. Also by `p; cs; cs0 and bs; we mean the spaces of absolutely p-summable, convergent, null and bounded series, respectively, where 1 ≤ p < 1: We write φ for the space of all sequences that terminate in zero. Moreover, we denote the space of all sequences of bounded variation by bv: A Banach space X is said to be a BK-space if it has continuous coordinates. The spaces `1; c0 and c are BK- spaces with norm kxk = sup jx j : Here and henceforth, for simplicity in notation, `1 k k the summation without limit runs from 0 to 1: Also, we shall use the notation e = (1; 1; 1;:::) and e(k) to be the sequence whose only non-zero term is 1 in the kth place for each k 2 N: Let X and Y be two sequence spaces and let A = (ank) be an infinite matrix of real th entries.
    [Show full text]
  • Chapter 4 the Lebesgue Spaces
    Chapter 4 The Lebesgue Spaces In this chapter we study Lp-integrable functions as a function space. Knowledge on functional analysis required for our study is briefly reviewed in the first two sections. In Section 1 the notions of normed and inner product spaces and their properties such as completeness, separability, the Heine-Borel property and espe- cially the so-called projection property are discussed. Section 2 is concerned with bounded linear functionals and the dual space of a normed space. The Lp-space is introduced in Section 3, where its completeness and various density assertions by simple or continuous functions are covered. The dual space of the Lp-space is determined in Section 4 where the key notion of uniform convexity is introduced and established for the Lp-spaces. Finally, we study strong and weak conver- gence of Lp-sequences respectively in Sections 5 and 6. Both are important for applications. 4.1 Normed Spaces In this and the next section we review essential elements of functional analysis that are relevant to our study of the Lp-spaces. You may look up any book on functional analysis or my notes on this subject attached in this webpage. Let X be a vector space over R. A norm on X is a map from X ! [0; 1) satisfying the following three \axioms": For 8x; y; z 2 X, (i) kxk ≥ 0 and is equal to 0 if and only if x = 0; (ii) kαxk = jαj kxk, 8α 2 R; and (iii) kx + yk ≤ kxk + kyk. The pair (X; k·k) is called a normed vector space or normed space for short.
    [Show full text]
  • Functional Analysis Lecture Notes Chapter 3. Banach
    FUNCTIONAL ANALYSIS LECTURE NOTES CHAPTER 3. BANACH SPACES CHRISTOPHER HEIL 1. Elementary Properties and Examples Notation 1.1. Throughout, F will denote either the real line R or the complex plane C. All vector spaces are assumed to be over the field F. Definition 1.2. Let X be a vector space over the field F. Then a semi-norm on X is a function k · k: X ! R such that (a) kxk ≥ 0 for all x 2 X, (b) kαxk = jαj kxk for all x 2 X and α 2 F, (c) Triangle Inequality: kx + yk ≤ kxk + kyk for all x, y 2 X. A norm on X is a semi-norm which also satisfies: (d) kxk = 0 =) x = 0. A vector space X together with a norm k · k is called a normed linear space, a normed vector space, or simply a normed space. Definition 1.3. Let I be a finite or countable index set (for example, I = f1; : : : ; Ng if finite, or I = N or Z if infinite). Let w : I ! [0; 1). Given a sequence of scalars x = (xi)i2I , set 1=p jx jp w(i)p ; 0 < p < 1; 8 i kxkp;w = > Xi2I <> sup jxij w(i); p = 1; i2I > where these quantities could be infinite.:> Then we set p `w(I) = x = (xi)i2I : kxkp < 1 : n o p p p We call `w(I) a weighted ` space, and often denote it just by `w (especially if I = N). If p p w(i) = 1 for all i, then we simply call this space ` (I) or ` and write k · kp instead of k · kp;w.
    [Show full text]
  • LEBESGUE MEASURE and L2 SPACE. Contents 1. Measure Spaces 1 2. Lebesgue Integration 2 3. L2 Space 4 Acknowledgments 9 References
    LEBESGUE MEASURE AND L2 SPACE. ANNIE WANG Abstract. This paper begins with an introduction to measure spaces and the Lebesgue theory of measure and integration. Several important theorems regarding the Lebesgue integral are then developed. Finally, we prove the completeness of the L2(µ) space and show that it is a metric space, and a Hilbert space. Contents 1. Measure Spaces 1 2. Lebesgue Integration 2 3. L2 Space 4 Acknowledgments 9 References 9 1. Measure Spaces Definition 1.1. Suppose X is a set. Then X is said to be a measure space if there exists a σ-ring M (that is, M is a nonempty family of subsets of X closed under countable unions and under complements)of subsets of X and a non-negative countably additive set function µ (called a measure) defined on M . If X 2 M, then X is said to be a measurable space. For example, let X = Rp, M the collection of Lebesgue-measurable subsets of Rp, and µ the Lebesgue measure. Another measure space can be found by taking X to be the set of all positive integers, M the collection of all subsets of X, and µ(E) the number of elements of E. We will be interested only in a special case of the measure, the Lebesgue measure. The Lebesgue measure allows us to extend the notions of length and volume to more complicated sets. Definition 1.2. Let Rp be a p-dimensional Euclidean space . We denote an interval p of R by the set of points x = (x1; :::; xp) such that (1.3) ai ≤ xi ≤ bi (i = 1; : : : ; p) Definition 1.4.
    [Show full text]
  • Linear Spaces
    Chapter 2 Linear Spaces Contents FieldofScalars ........................................ ............. 2.2 VectorSpaces ........................................ .............. 2.3 Subspaces .......................................... .............. 2.5 Sumofsubsets........................................ .............. 2.5 Linearcombinations..................................... .............. 2.6 Linearindependence................................... ................ 2.7 BasisandDimension ..................................... ............. 2.7 Convexity ............................................ ............ 2.8 Normedlinearspaces ................................... ............... 2.9 The `p and Lp spaces ............................................. 2.10 Topologicalconcepts ................................... ............... 2.12 Opensets ............................................ ............ 2.13 Closedsets........................................... ............. 2.14 Boundedsets......................................... .............. 2.15 Convergence of sequences . ................... 2.16 Series .............................................. ............ 2.17 Cauchysequences .................................... ................ 2.18 Banachspaces....................................... ............... 2.19 Completesubsets ....................................... ............. 2.19 Transformations ...................................... ............... 2.21 Lineartransformations.................................. ................ 2.21
    [Show full text]
  • 10. Uniformly Convex Spaces
    Researcher, 1(1), 2009, http://www.sciencepub.net, [email protected] Uniformly Convex Spaces Usman, M .A, Hammed, F.A. And Olayiwola, M.O Department Of Mathematical Sciences Olabisi Onabanjo University Ago-Iwoye, Nigeria [email protected] ABSTARACT: This paper depicts the class OF Banach Spaces with normal structure and that they are generally referred to as uniformly convex spaces. A review of properties of a uniformly convex spaces is also examined with a view to show that all non-expansive mapping have a fixed point on this space. With the definition of uniformly convex spaces in mind, we also proved that some spaces are uniformly spaces. [Researcher. 2009;1(1):74-85]. (ISSN: 1553-9865). Keywords: Banach Spaces, Convex spaces, Uniformly Convex Space. INTRODUCTION The importance of Uniformly convex spaces in Applied Mathematics and Functional Analysis, it has developed into area of independent research, where several areas of Mathematics such as Homology theory, Degree theory and Differential Geometry have come to play a very significant role. [1,3,4] Classes of Banach spaces with normal structure are those generally refer to as Uniformly convex spaces. In this paper, we review properties of the space and show that all non- expansive maps have a fixed-point on this space. [2] Let x be a Banach Space. A Branch space x is said to be Uniformly convex if for ε > 0 there exist a ∂ = (ε ) > 0 such that if x, y £ x with //x//=1, //y//=1 and //x-y// ≥ ε , then 1 ()x+ y // ≤ 1 − ∂ // 2 . THEOREM (1.0) Let x = LP (µ) denote the space of measurable function f such that //f// are integrable, endowed with the norm.
    [Show full text]
  • ON the STABILITY of the DIFFERENTIAL PROCESS GENERATED by COMPLEX INTERPOLATION Contents 1. Introduction 2 2. Preliminary Result
    ON THE STABILITY OF THE DIFFERENTIAL PROCESS GENERATED BY COMPLEX INTERPOLATION JESUS´ M. F. CASTILLO, WILLIAN H. G. CORREA,^ VALENTIN FERENCZI, MANUEL GONZALEZ´ Abstract. We study the stability of the differential process of Rochberg and Weiss associated to an analytic family of Banach spaces obtained using the complex interpolation method for families. In the context of K¨othefunction spaces we complete earlier results of Kalton (who showed that there is global bounded stability for pairs of K¨othespaces) by showing that there is global (bounded) stability for families of up to three K¨othespaces distributed in arcs on the unit sphere while there is no (bounded) stability for families of four or more K¨othespaces. In the context of arbitrary pairs of Banach spaces we present some local stability results and some global isometric stability results. Contents 1. Introduction 2 2. Preliminary results 4 3. Kalton spaces of analytic functions 6 3.1. Derivations, centralizers and twisted sums 6 3.2. Distances and isomorphisms 8 3.3. Bounded stability 10 4. Stability of splitting for K¨othefunction spaces 11 4.1. Stability for pairs of K¨othespaces 12 4.2. Complex interpolation for families 15 4.3. Stability for families of three K¨othespaces 17 4.4. (Bounded) stability fails for families of four K¨othespaces 22 5. Stability of splitting for pairs of Banach spaces 25 5.1. Local bounded stability for coherent pairs 26 5.2. Isometric rigidity of linear derivations for optimal interpolation pairs 28 References 30 2010 Mathematics Subject Classification. 46B70, 46E30, 46M18. Key words and phrases.
    [Show full text]
  • Multiplication and Composition Operators on Weak Lp Spaces
    Multiplication and Composition Operators on Weak Lp spaces Ren´eErlin Castillo1, Fabio Andr´esVallejo Narvaez2 and Julio C. Ramos Fern´andez3 Abstract In a self-contained presentation, we discuss the Weak Lp spaces. Invertible and compact multiplication operators on Weak Lp are char- acterized. Boundedness of the composition operator on Weak Lp is also characterized. Keywords: Compact operator, multiplication and composition oper- ator, distribution function, Weak Lp spaces. Contents 1 Introduction 2 2 Weak Lp spaces 2 3 Convergence in measure 7 4 An interpolation result 13 5 Normability of Weak Lp for p > 1 25 6 Multiplication Operators 39 7 Composition Operator 49 A Appendix 52 02010 Mathematics subjects classification primary 47B33, 47B38; secondary 46E30 1 1 Introduction One of the real attraction of Weak Lp space is that the subject is sufficiently concrete and yet the spaces have fine structure of importance for applications. WeakLp spaces are function spaces which are closely related to Lp spaces. We do not know the exact origin of Weak Lp spaces, which is a apparently part of the folklore. The Book by Colin Benett and Robert Sharpley[4] contains a good presentation of Weak Lp but from the point of view of rearrangement function. In the present paper we study the Weak Lp space from the point of view of distribution function. This circumstance motivated us to undertake a preparation of the present paper containing a detailed exposition of these function spaces. In section 6 of the present paper we first prove a charac- terization of the boundedness of Mu in terms of u, and show that the set of multiplication operators on Weak Lp is a maximal abelian subalgebra of B WeakLp , the Banach algebra of all bounded linear operators on WeakLp.
    [Show full text]
  • Lp Spaces in Isabelle
    Lp spaces in Isabelle Sebastien Gouezel Abstract Lp is the space of functions whose p-th power is integrable. It is one of the most fundamental Banach spaces that is used in analysis and probability. We develop a framework for function spaces, and then im- plement the Lp spaces in this framework using the existing integration theory in Isabelle/HOL. Our development contains most fundamental properties of Lp spaces, notably the Hölder and Minkowski inequalities, completeness of Lp, duality, stability under almost sure convergence, multiplication of functions in Lp and Lq, stability under conditional expectation. Contents 1 Functions as a real vector space 2 2 Quasinorms on function spaces 4 2.1 Definition of quasinorms ..................... 5 2.2 The space and the zero space of a quasinorm ......... 7 2.3 An example: the ambient norm in a normed vector space .. 9 2.4 An example: the space of bounded continuous functions from a topological space to a normed real vector space ....... 10 2.5 Continuous inclusions between functional spaces ....... 10 2.6 The intersection and the sum of two functional spaces .... 12 2.7 Topology ............................. 14 3 Conjugate exponents 16 4 Convexity inequalities and integration 18 5 Lp spaces 19 5.1 L1 ................................. 20 5.2 Lp for 0 < p < 1 ......................... 21 5.3 Specialization to L1 ....................... 22 5.4 L0 ................................. 23 5.5 Basic results on Lp for general p ................ 24 5.6 Lp versions of the main theorems in integration theory .... 25 1 5.7 Completeness of Lp ........................ 26 5.8 Multiplication of functions, duality ............... 26 5.9 Conditional expectations and Lp ...............
    [Show full text]
  • 3. the Lp Spaces (1 ≤ P < ∞)
    3. The Lp spaces (1 ≤ p < ∞) In this section we discuss an important construction, which is extremely useful in virtually all branches of Analysis. In Section 1, we have already introduced the space L1. The first construction deals with a generalization of this space. Definitions. Let (X, A, µ) be a measure space, and let K be one of the fields R or C. A. For a number p ∈ (1, ∞), we define the space Z Lp (X, A, µ) = f : X → : f measurable, and |f|p ∈ dµ < ∞ . K K X R Here we use the convention introduced in Section 1, which defines X h dµ = ∞, for those measurable functions h : X → [0, ∞], that are not integrable. Of course, in this definition we can allow also the value p = 1, and in this case we get the familiar definition of L1 (X, A, µ). K B. For p ∈ [1, ∞), we define the map Q : L1 (X, A, µ) → [0, ∞) by p K Z Q (f) = |f|p dµ, ∀ f ∈ L1 (X, A, µ). p K X Remark 3.1. The space L1 (X, A, µ) was studied earlier (see Section 1). It K has the following features: (i) L1 (X, A, µ) is a -vector space. K K (ii) The map Q : L1 (X, A, µ) → [0, ∞) is a seminorm, i.e. 1 K (a) Q (f + g) ≤ Q (f) + Q (g), ∀ f, g ∈ L1 (X, A, µ); 1 1 1 K (b) Q (αf) = |α| · Q (f), ∀ f ∈ L1 (X, A, µ), α ∈ . 1 1 K K (iii) R f dµ ≤ Q (f), ∀ f ∈ L1 (X, A, µ).
    [Show full text]
  • Some Types of Banach Spaces, Hermitian Operators, And
    SOME TYPES OF BANACHSPACES, HERMITIAN OPERATORS,AND BADE FUNCTIONAL^1) BY EARL BERKSON Introduction. In [6] and [ll] a general notion of hermitian operator has been developed for arbitrary complex Banach spaces (see §1 below). In terms of this notion, a family of operators on a Banach space is said to be hermitian-equivalent if the operators of this family can be made simultaneously hermitian by equivalent renorming of the underlying space [7]. Let X be a complex Banach space with norm || ||, and let F be a commutative hermitian-equivalent family of operators on X. A norm for X equivalent to || ||, and relative to which the operators of F are hermitian will be called an F-norm. Such families have been studied in [7], where it is shown that if X is a Hilbert space, then there is an F-norm which is also a Hilbert space norm. It is natural to seek other properties which, if enjoyed by X, can be preserved by choosing an F-norm appro- priately. Such an investigation is conducted in this paper. Specifically, we show in §§3 and 4 that if the Banach space X has uniformly Frechet differentiable norm (resp., is uniformly convex), then there is an F-norm which preserves uniform Frechet differentiability (resp., uniform con- vexity). Moreover, we show in §6 that if X = Lp(fi), <*>> p > 1, M a meas- ure, then there is an F-norm which preserves both uniform Frechet differ- entiability and uniform convexity. Our result for'the case where X is uniformly convex enables us to es- tablish in Theorem (5.4) a strong link between the notions of semi-inner- product (see §1) and Bade functional.
    [Show full text]