Lp-Structure in Real Banach Spaces

Behrends, Ehrhard; Danckwerts, Rainer; Evans, Richard; Göbel, Silke M.; Greim, Peter; Meyfarth, Konrad; Müller, Winfried

doi:10.1007/bfb0068175

Cited by 34 publications

(36 citation statements)

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“…Therefore, we deduce that M = H, and the only thing we have to prove is that L has a maximal element. 2 , then we will have proved that cl (M ) ∈ L, and thus cl (M ) is an upper bound for the chain (M α ) α∈I . In that case, Zorn's lemma allows us to deduce the existence of maximal elements in L. Taking limits, it is pretty easy to show that cl (M ) ∩ L = {0}.…”

Section: Theorem 41 Let X Be a Real Banach Space Let U Be An Lmentioning

confidence: 99%

“…Assume that (1) holds. In order to show (2), it suffices to prove that J X (x) = J X (m) + δJ X (e) for every x = m + δe ∈ X (where m ∈ ker (J X (e)) and δ ∈ R). Pick n ∈ ker (J X (e)) and λ ∈ R. By Hölder's inequality,…”

Section: The Particular Case Of Smooth Spacesmentioning

confidence: 99%

“…Notice that N is uniquely determined, and hence so is π M . Good references for L 2 -summand subspaces are [2] and [3].…”

Section: Introductionmentioning

confidence: 99%

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𝖫^{2}-summand vectors in Banach spaces

Aizpuru¹,

García-Pacheco

2006

Proc. Amer. Math. Soc.

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Abstract. The aim of this paper is to study the set L 2 X of all L 2 -summand vectors of a real Banach space X. We provide a characterization of L 2 -summand vectors in smooth real Banach spaces and a general decomposition theorem which shows that every real Banach space can be decomposed as an L 2 -sum of a Hilbert space and a Banach space without nontrivial L 2 -summand vectors. As a consequence, we generalize some results and we obtain intrinsic characterizations of real Hilbert spaces. BackgroundWe say that a Banach space X is smooth if every point x in X satisfies that there exists a unique f in X * such that f = x and f (x) = x 2 . If X is a smooth Banach space and x is in X, then we will denote this f by J X (x). The mapping J X : X −→ X * is usually called the duality mapping. Excellent books for consulting the duality mapping are [7] and [8].A closed subspace M of a real Banach space X is said to be an L 2 -summand subspace if there exists another closed subspace N of X verifying X = (M ⊕ N ) 2 ; in other words, m + n 2 = m 2 + n 2 for every m in M and every n in N . The linear projection π M of X onto M that fixes the elements of M and maps the elements of N to {0} is called the L 2 -summand projection of X onto M . Notice that N is uniquely determined, and hence so is π M . Good references for L 2 -summand subspaces are [2] and [3].A vector e of a real Banach space X is an L 2 -summand vector if Re is an L 2 -summand subspace. Furthermore, if e = 0, then there exists a functional e * in X * , which is called the L 2 -summand functional of e, such that e * = 1/ e , e * (e) = 1 and π Re (x) = e * (x) e for every x in X. We want to recall two relevant results about L 2 -summand vectors:(1

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Section: Theorem 41 Let X Be a Real Banach Space Let U Be An Lmentioning

confidence: 99%

Section: The Particular Case Of Smooth Spacesmentioning

confidence: 99%

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𝖫^{2}-summand vectors in Banach spaces

Aizpuru¹,

García-Pacheco

2006

Proc. Amer. Math. Soc.

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“…= 1 the necessity is easily seen, since (1) implies that the support of x is an atom (see also [6]). …”

mentioning

confidence: 99%

“…We give an example of a nonseparable Banach space V and a function x on [0,1] with values in the unit sphere of V that is an extreme point of the unit balls of all Bochner L'-spaces Lp(\, V), 1 < p < oo, a Lebesgue measure, though none of its values is an extreme point of the unit ball of V. This shows that a characterization of the extremal elements in Lp(k, V) for separable V, given by J. A. Johnson, does not hold in general.…”

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confidence: 99%

An extremal vector-valued 𝐿^{𝑝}-function taking no extremal vectors as values

Greim¹

1982

Proc. Amer. Math. Soc.

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Abstract. We give an example of a nonseparable Banach space V and a function x on [0,1] with values in the unit sphere of V that is an extreme point of the unit balls of all Bochner L'-spaces Lp(\, V), 1 < p < oo, a Lebesgue measure, though none of its values is an extreme point of the unit ball of V. This shows that a characterization of the extremal elements in Lp(k, V) for separable V, given by J. A. Johnson, does not hold in general.The extremal elements in the unit sphere of vector-valued CK-or TAspaces have been studied by many authors, e.g. in [2, 5 and 6]. (For the definition and elementary properties of Bochner L^-spaces we refer the reader to [4].) A quite natural question is to ask whether such a function x is extremal if and only if(1) the function ||x(-)|| is extremal in the corresponding scalar function space and (2) the vector x(t) is extremal in the ball with radius ||x(i)|| for all t in a dense subset of the base space (in the CK-case) resp. for almost all t (in the TZ-case).The "if" part is easy and well known; on the other hand, necessity of (1) is trivial. Hence the remaining question is the necessity of (2).In the case/? = 1 the necessity is easily seen, since (1) implies that the support of x is an atom (see also [6]).The CK-case was settled long ago. Blumenthal, Lindenstrauss, and Phelps have shown in [2] that for real range spaces V with dimension < 3 the condition (2) is necessary. On the other hand they give an example of a 4-dimensional space V and an extremal x in C([0, 1], V) taking no extremal values. In the remaining cases 1

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