Hahn-Banach operators
22 Feb 2001-Vol. 129, Iss: 10, pp 2923-2930
TL;DR: In this paper, Buskes et al. considered real linear spaces and showed that for every isometric embedding of the space X into a Banach space Z, there exists a norm-preserving extension T of T to Z. The latter result is a generalization of a recent result due to B. Chalmers and B. Shekhtman.
Abstract: We consider real spaces only. Definition. An operator T : X → Y between Banach spaces X and Y is called a Hahn-Banach operator if for every isometric embedding of the space X into a Banach space Z there exists a norm-preserving extension T of T to Z. A geometric property of Hahn-Banach operators of finite rank acting between finite-dimensional normed spaces is found. This property is used to characterize pairs of finite-dimensional normed spaces (X, Y ) such that there exists a Hahn-Banach operator T : X → Y of rank k. The latter result is a generalization of a recent result due to B. L. Chalmers and B. Shekhtman. Everywhere in this paper we consider only real linear spaces. Our starting point is the classical Hahn-Banach theorem ([H], [B1]). The form of the Hahn-Banach theorem we are interested in can be stated in the following way. Hahn-Banach Theorem. Let X and Y be Banach spaces, T : X → Y a bounded linear operator of rank 1 and Z a Banach space containing X as a subspace. Then there exists a bounded linear operator T : Z → Y satisfying (a) ||T || = ||T ||; (b) T x = Tx for every x ∈ X. Definition 1. An operator T : Z → Y satisfying (a) and (b) for a bounded linear operator T : X → Y is called a norm-preserving extension of T to Z. The Hahn-Banach theorem is one of the basic principles of linear analysis. It is quite natural that there exists a vast literature on generalizations of the HahnBanach theorem for operators of higher rank. See papers by G. P. Akilov [A], J. M. Borwein [Bor], B. L. Chalmers and B. Shekhtman [CS], G. Elliott and I. Halperin [EH], D. B. Goodner [Go], A. D. Ioffe [I], S. Kakutani [Kak], J. L. Kelley [Kel], J. Lindenstrauss [L1], [L2], L. Nachbin [N1] and M. I. Ostrovskii [O], representing different directions of such generalizations, and references therein. There exist two interesting surveys devoted to the Hahn-Banach theorem and its generalizations; see G. Buskes [Bus] and L. Nachbin [N2]. We shall use the following natural definition. Definition 2. An operator T : X → Y between Banach spaces X and Y is called a Hahn-Banach operator if for every isometric embedding of the space X into a Banach space Z there exists a norm-preserving extension T of T to Z. Received by the editors February 9, 2000. 2000 Mathematics Subject Classification. Primary 46B20, 47A20.
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01 Mar 2007
TL;DR: The Hahn-Banach theorem as discussed by the authors states that a continuous linear functional on a subspace M of a normed space X can be extended to a functional on X without changing its norm.
Abstract: I love the Hahn-Banach theorem. I love it the way I love Casablanca and the Fontana di Trevi. It is something not so much to be read as fondled. What is “the Hahn-Banach theorem?” Let f be a continuous linear functional defined on a subspace M of a normed space X. Take as the Hahn-Banach theorem the property that f can be extended to a continuous linear functional on X without changing its norm. Innocent enough, but the ramifications of the theorem pervade functional analysis and other disciplines (even thermodynamics!) as well. Where did it come from? Were Hahn and Banach the discoverers? The axiom of choice implies it, but what about the converse? Is Hahn-Banach equivalent to the axiom of choice? (No.) Are Hahn-Banach extensions ever unique? They are in more cases than you might think, when the unit ball of the dual is “round,” as for `p with 1 1, for 1 < p < ∞, despite the topologies being identical. The cubic nature of the unit ball does not suffice, however—if Y = c0, the extendibility dies. This article traces the evolution of the analytic form as well as subsequent developments up to 2004.
8 citations
01 Jan 1987
TL;DR: In this article, the summing norms are estimated using a restricted number of elements, and the interpolation method for Grothendieck-type theorems is used.
Abstract: 0. Banach space background 1. Finite rank operators: trace and 1-nuclear norm 2. Finite sequences of elements : the quantities 1, 2 3. The summing norms 4. Other nuclear norms: duality with the summing norms 5. Pietsch's theorem and its applications 6. Averaging: type 2 and cotype 2 constants 7. More averaging: Khinchin's inequality and related results 8. Integral methods: Gaussian averaging 9. 2-dominated spaces 10. Grothendieck's inequality 11. The interpolation method for Grothendieck-type theorems 12. Results connected with the basis constant 13. Estimation of summing norms using a restricted number of elements 14. Pisier's theorem for pi2,1 15. Tensor products of operators 16. Trace duality revisited: integral norms 17. Applications of local reflexivity 18. Cone-summing norms.
5 citations
TL;DR: Yilmaz et al. as mentioned in this paper introduced the notion of relative adjoint operators and characterized some operator spaces by this notion and by the results presented in [Y.Y.C., 2004], where they proved that the operator space L ( l ∞ ( A, X ), c 0 ( A, Z ) ) is equivalent to c 0( A, L SOT ( l∞ ( X, Z ), Z ) in the sense of isometric isomorphism.
Abstract: In this work, we introduce the notion of relative adjoint operators and characterize some operator spaces by this notion and by the results presented in [Y. Yilmaz, Structural properties of some function spaces, Nonlinear Anal. 59 (2004) 959–971]. Hence, for example, we prove that the operator space L ( l ∞ ( A , X ) , c 0 ( A , Z ) ) is equivalent to c 0 ( A , L SOT ( l ∞ ( A , X ) , Z ) ) in the sense of isometric isomorphism, where A is an infinite set, X , Z are Banach spaces and L SOT ( X , Z ) is the space L ( X , Z ) endowed with the strong operator topology. Note that the vector-valued function spaces l ∞ ( A , X ) and c 0 ( A , Z ) , defined in the prerequisites, are important generalizations of the classical Banach spaces l ∞ and c 0 .
4 citations
TL;DR: In this paper, the authors review ongoing work on operators having norm-preserving extensions to every overspace. They call them Hahn-Banach operators and refer to them as Hahn operators.
Abstract: In this paper we review ongoing work on operators having norm-preserving extensions to every overspace. We call them Hahn-Banach operators.
2 citations
TL;DR: In this article, the space of vector valued multipliers of strongly Henstock-Kurzweil integrable functions is investigated and it is shown that if X is a commutative Banach algebra with identity e such that e = 1 and g : [a, b] → X is of strongly bounded variation, then the multiplication operator defined by Mg(f) := fg maps 𝒮ℋ𝒦 to ℋ ǫ.
Abstract: Abstract We investigate the space of vector valued multipliers of strongly Henstock-Kurzweil integrable functions. We prove that if X is a commutative Banach algebra with identity e such that ‖e‖ = 1 and g : [a, b] → X is of strongly bounded variation, then the multiplication operator defined by Mg(f) := fg maps 𝒮ℋ𝒦 to ℋ𝒦. We also prove a partial converse, when X is a Gel’fand space.
1 citations
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01 Feb 1993TL;DR: Inequalities for mixed volumes 7. Selected applications Appendix as discussed by the authors ] is a survey of mixed volumes with bounding boxes and quermass integrals, as well as a discussion of their applications.
Abstract: 1. Basic convexity 2. Boundary structure 3. Minkowski addition 4. Curvature measure and quermass integrals 5. Mixed volumes 6. Inequalities for mixed volumes 7. Selected applications Appendix.
3,954 citations
TL;DR: In this paper, the authors give a characterization of linear and completely continuous transformations both on the common Banach spaces to an arbitrary Banach space and vice versa, and restate a few of the results obtained by Gelfand [12], and Gowurin [13 ].
Abstract: The purpose of this paper is to give a characterization of linear and completely continuous transformations both on the common Banach spaces to an arbitrary Banach space and vice versa. There is an abundant literature on this subject. Among the earliest papers, the now famous paper of Radon [241 should be mentioned. Here linear transformations on LP to Lq (1
214 citations
TL;DR: In this article, a necessary and sufficient condition for the space where the transformation takes its values and does not involve the transformation itself or space where it is to be defined, for such an extension to be possible is expressed in terms of a certain "binary intersection property" of the collection of spheres of the normed space.
Abstract: Every continuous linear functional defined on a vector subspace of a real normed space can be extended to the whole space so as to remain linear and continuous, and with the same norm(2). The extension of continuous linear transformations between two real normed spaces has been studied by several authors and for a long time it has been recognized that this problem has a close connection with the question of the existence of projections of norm one, and moreover that the nature of the space where the transformations take their values is much more important than that of the space where the transformations have to be defined. It is not known, as far as we can say, what are the precise conditions for the possibility of extending a transformation without disturbing its linearity, continuity, and norm. In this paper we shall give a necessary and sufficient condition, which refers only to the space where the transformation takes its values and does not involve the transformation itself or the space where it is to be defined, for such an extension to be possible: the condition is expressed in terms of a certain "binary intersection property" of the collection of spheres of the normed space (see Theorem 1). After that we proceed to the study of the structure of real normed spaces whose collections of spheres have this property. A first step in this direction is given by the theorem asserting that these normed spaces (provided they contain at least one extreme point in the unity sphere) are simply those that can be made into complete vector lattices with order unity in such a manner that the norm derived from the order relation and the order unity in the natural way is identical to the given norm (see Theorem 2, and Theorem 3 for the finite-dimensional case). In this connection we point out a conjecture which we have not been able to settle, namely that, if the collection of spheres of a normed space has the binary intersection property, then its unity sphere must contain an extreme point. By using some results of S. Kakutani and M. H. Stone, we establish the connection between the normed spaces having the binary intersection property and the spaces of real continuous functions over certain compact Hausdorff spaces (see Theorem 4), or the complete Boolean algebras (see Theorem 6).
189 citations
TL;DR: Goodner and Nachbin this article showed that if B has the extension property and if its unit sphere has an extreme point, then B is equivalent to a function space of this sort.
Abstract: Recently, in these Transactions, Nachbin [N] and, independently, Goodner [G] have shown that if B has the extension property and if its unit sphere has an extreme point, then B is equivalent to a function space of this sort; both authors have also proved that such a function space has the extension property The above theorem simply omits the extreme point hypothesis, and so establishes the equivalence My original proof, of which the proof given here is a distillate, depends on an idea of Jerison [j] Briefly, letting X be the weak* closure of the set of extreme points of the unit sphere of the adjoint B*, B can be shown equivalent to the space of all weak* continuous real functions / on X such that/(x) = —f( — x), and then properties of X are deduced which imply the theorem The same idea occurs implicitly in the proof below Note Goodner asks [G, p 107] if every Banach space having the extension property is equivalent to the conjugate of an abstract (L)-space It is known (this is not my contribution) that the Birkhoff-Ulam example ([B, p 186] or [HT, p 490]) answers this question in the negative, the pertinent Banach space being the bounded Borel functions on [0, 1 ] modulo those functions vanishing except on a set of the first category, with ||/|| = inf {K: \f(x) | g K save on a set of first category} 1 Preliminary definitions and remarks A point x is an extreme point of a convex subset K of a real linear space if x is not an interior point of any line segment contained in K (ie, if x=ty + (l — t)z, 0
146 citations