A multilinear Lindenstrauss theorem (original) (raw)

A Lindenstrauss theorem for some classes of multilinear mappings

Journal of Mathematical Analysis and Applications, 2014

Under some natural hypotheses, we show that if a multilinear mapping belongs to some Banach multlinear ideal, then it can be approximated by multilinear mappings belonging to the same ideal whose Arens extensions simultaneously attain their norms. We also consider the class of symmetric multilinear mappings.

Denseness of norm attaining mappings

Revista de la Real Academia de Ciencias …, 2006

The Bishop-Phelps Theorem states that the set of (bounded and linear) functionals on a Banach space that attain their norms is dense in the dual. In the complex case, Lomonosov proved that there may be a closed, convex and bounded subset C of a Banach space such that the set of functionals whose maximum modulus is attained on C is not dense in the dual. This paper contains a survey of versions for operators, multilinear forms and polynomials of the Bishop-Phelps Theorem. Lindenstrauss provided examples of Banach spaces X and Y such that the set of norm attaining operators from X to Y is not dense. He also gave isometric conditions on X for which the set of norm attaining operators from X to Y are dense in the space of all operators between these Banach spaces. If the above conclusion holds for every Y , X is said to have the property A. Also, there are known sufficient conditions on the range space Y in order to have the same denseness conditions for every X. In such a case, Y has the property B. Bourgain proved that every space satisfying the Radon-Nikodým property has the property A. For classical Banach spaces, it is known that C[0, 1], p (1 < p < ∞) and any infinite-dimensional L1(µ) do not satisfy the property B (results due to Schachermayer, Gowers and Acosta, respectively). When both X and Y are either C(K) or L 1 (µ), there are positive results due to Johnson and Wolfe, and Iwanik, respectively. Finet and Payá proved that there is also positive result for X = L1 and Y = L∞. For multilinear mappings, Aron, Finet and Werner initiated the research and gave sufficient conditions on a Banach space X in order to satisfy the denseness of the set of norm attaining N-linear mappings in the set of all the N-linear mappings (Radon-Nikodým property, for instance). Choi showed that the space L 1 [0, 1] does not satisfy the denseness of the set of norm attaining bilinear forms. Alaminos, Choi, Kim and Payá proved that for any scattered compact space K, the set of norm attaining N-linear forms on C(K) is dense in the space of all N-linear forms, and for the bilinear case no restriction on the compact is needed. Acosta, García and Maestre proved that the set of N-linear forms whose Arens extensions to the bidual attains the norm is dense in the space of all the N-linear forms on a product of N Banach spaces. For polynomials and for holomorphic mappings, there are some results along the same line, but more open problems than for the multilinear case. Densidad de funciones que alcanzan la norma Resumen. El Teorema de Bishop-Phelps afirma que para cualquier espacio de Banach X, el conjunto de los funcionales (lineales y continuos) que alcanzan la norma es denso en el dual. En el caso complejo, Lomonosov dio ejemplos de conjuntos convexos, cerrados y acotados C, tales que el conjunto de los funcionales cuyo máximo se alcanza en C no es denso en el dual. Este trabajo contiene diversas versiones del Teorema de Bishop-Phelps para operadores, formas multilineales y polinomios. Lindenstrauss dio Presentado por Vicente Montesinos Santalucía.

On the Lindenstrauss-Rosenthal theorem

Israel Journal of Mathematics, 2004

We present a homological principle that governs the behaviour of couples of exact sequences of quasi-Banach spaces. Three applications are given: (i) A unifying method of proof for the results of Lindenstrauss, Rosenthal, Kalton, Peck and Kislyakov about the extension and lifting of isomorphisms in co, loo, lp and Lp for 0 < p < 1; (ii) A study of the Dunford-Pettis property in duals of quotients of £oo-spaces; and (iii) New results on the extension of C(K)-valued operators. Isr. J. Math. PROPOSITION 1.3 (case /oo): Let i and j be two injective isomorphisms from a Banach space Y into loo in such a way that both l~/iY and l~/jY are not reflexive. Then there is an automorphism r of l~ such that 7i = j. If the quotients loo/iY and loo/jY are both reflexive, then the automorphism T exists if and only if the Fredholm index of any extension of j to loo through i is O. If one of the quotients is reflexive but the other is not, no such automorphism exists. Kalton extended in [16] the first of those results to/p-spaces for 0 < p < 1: PROPOSITION 1.4 (case /;): Let 0 < p < 1 and let q and Q be two quotient maps onto a quasi-Banach space X not isomorphic to Ip. J[f kerq and ker Q contain copies of lp complemented in lp, then there exists an automorphism T Of lp SUCh that qT -~ Q. In [18] Kalton and Peck obtained some variations of this result for Lp(O, 1), 0~p<l. PROPOSITION 1.5 (case Lp): Let 0 < p < 1 and let q and Q be two quotient maps onto a quasi-Banach space X in such a way that ker q and ker Q are either q-Banach spaces for some q > p or ultrasummand spaces. Then there exists an automorphism T of Lp such that qv = Q. In [21] Kislyakov considered the case L1, obtaining: PROPOSITION 1.6 (case LI): Let A and B be two reflexive subspaces of LI(#) such that LI(p)/A = LI(p)/B. Then one of the subspaces A, B is isomorphic to the product of the other one with a finite dimensional space Lindenstrauss showed in [23] a partial converse of the/1-result and applied it to solve a problem raised in [26] about the existence of infinitely many isomorphy types of £1-spaces. PROPOSITION 1.7 (Ex subspaces of/1): Let A,B be two f~l-spaces, and let qA: l~ -+ A and qB: ll --+ B be two quotient operators with infinite-dimensional kernels. Then ker qA is isomorphic to ker qB if and only if A and B are isomor~ phic.

Norm Attaining Arens Extensions on

Abstract and Applied Analysis, 2014

We study norm attaining properties of the Arens extensions of multilinear forms defined on Banach spaces. Among other related results, we construct a multilinear form onℓ1with the property that only some fixed Arens extensions determined a priori attain their norms. We also study when multilinear forms can be approximated by ones with the property that only some of their Arens extensions attain their norms.

On n-normed spaces

International Journal of Mathematics and Mathematical Sciences, 2001

Abstract. Given an n-normed space with n ≥ 2, we offer a simple way to derive an (n−1)- norm from the n-norm and realize that any n-normed space is an (n−1)-normed space. We also show that, in certain cases, the (n−1)-norm can be derived from the n-norm in such a way that ...

On nnn-norms and bounded nnn-linear functionals in a hilbert space

Annals of Functional Analysis, 2010

In this paper we discuss the concept of n-normed spaces. In particular, we show the equality of four different formulas of n-norms in a Hilbert space. In addition, we study the notion of bounded n-linear functionals on an n-normed space and present some results on it.

A Two-Dimensional Hahn-Banach Theorem

Proceedings of the American Mathematical Society

be an extension of T to all of X (i.e., u i ∈ X * ) such that T has minimal (operator) norm. In this paper we show in particular that, in the case n = 2 and the field is R, there exists a rank-nT such that T = T for all X if and only if the unit ball of V is either not smooth or not strictly convex. In this case we show, furthermore, that, for some T = T , there exists a choice of basis v = v 1 , v 2 such that u i = ũ i , i = 1, 2; i.e., each u i is a Hahn-Banach extension ofũ i .

A structural version of the theorem of Hahn-Banach

We consider one of the basic results of functional analysis, the classical theorem of Hahn-Banach. This theorem gives the existence of a continuous linear functional on a given normed vectorspace extending a given continuous linear functional on a subspace with the same norm. In this paper we generalize this existence theorem to a result on the structure of the set of all these extensions.

Extensions of multilinear operators and Banach space properties

Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 2003

A new characterization of the Dunford-Pettis property in terms of the extensions of multilinear operators to the biduals is obtained. For the first time, multilinear characterizations of the reciprocal Dunford-Pettis property and Pe lczyński's property (V) are also found. Polynomial and holomorphic versions of these properties are given as well.

The Product - Normed Linear Space

International Journal for Research in Applied Science and Engineering Technology, 2019

In this paper, The combination of the structure of a vector space with the structure of a metric space naturally produces the structure of a normed space and a Banach space, i.e., of a complete linear normed space. The abstract definition of a linear normed space first appears around 1920 in the works of Stefan Banach (1892-1945), Hans Hahn (1879-1934) and Norbert Wiener (1894-1964). In fact, it is in these years that the Polish school around Banach discovered the principles and laid the foundation of what we now call linear functional analysis.