Extension of polynomials defined on subspaces

Fernández-Unzueta, Maite; Prieto, Ángeles

doi:10.1017/s0305004110000022

Cited by 9 publications

(14 citation statements)

References 21 publications

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“…Theorem 26. ( [5], see also [6]): Let X be a Hilbert space. Then for each subspace Y of X, there exists a bounded linear 'extension' map F : Y * → X * , i.e, F (g)| Y = g, g ∈ Y * .…”

Section: Problem 3: Does the Converse Of The Previous Statement Hold?mentioning

confidence: 99%

Banach limits: some new thoughts and perspectives

Sofi

2019

J Anal

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The existence of a Banach limit as a translation invariant positive continuous linear functional on the space of bounded scalar sequences which is equal to 1 at the constant sequence (1, 1, . . . , 1, . . .) is proved in a first course on functional analysis as a consequence of the Hahn Banach extension theorem. Whereas its use as an important tool in classical summability theory together with its application in the existence of certain invariant measures on compact (metric) spaces is well known, a renewed interest in the theory of Banach limits has led to certain applications which have opened new vistas in the structure of Banach spaces.The paper is devoted to a discussion of certain developments, both classical and recent, surrounding the theory of Banach limits including the structure of the set of Banach limits with special emphasis on certain aspects of their applications to the existence of certain invariant measures, vector valued analogues of Banach limits, functional equations and in the structure theory of Banach spaces involving the existence of selectors of certain multi-valued mappings into the metric space of non-empty, convex, closed and bounded subsets of a Banach space with respect to the Hausdorff metric. The paper shall conclude with a brief description of some recent results of the author on the study of 'simultaneous continuous linear' operators (linear selections) involving Hahn Banach extensions on spaces of Lipschitz functions on (subspaces of) Banach spaces.1.1. Existence of Banach limits. One of the common methods to prove the existence of Banach limits is via the use of the invariant version of the Hahn Banach theorem whereas the first proof by M.M. 0

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“…Theorem 26. ( [5], see also [6]): Let X be a Hilbert space. Then for each subspace Y of X, there exists a bounded linear 'extension' map F : Y * → X * , i.e, F (g)| Y = g, g ∈ Y * .…”

Section: Problem 3: Does the Converse Of The Previous Statement Hold?mentioning

confidence: 99%

Banach limits: some new thoughts and perspectives

Sofi

2019

J Anal

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“…( * )There exists c > 0 such that for each finite dimensional subspace Y of X, there exists a continuous linear extension map ψ : Y * → X * with ψ ≤ c (See [13] for details).…”

Section: Hilbert Spaces Determined Via Hahn Banach Phenomenamentioning

confidence: 99%

Some problems in functional analysis inspired by Hahn-Banach type theorems

Sofi¹

2014

Ann. Funct. Anal.

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Abstract. As a cornerstone of functional analysis, Hahn-Banach theorem constitutes an indispensable tool of modern analysis where its impact extends beyond the frontiers of linear functional analysis into several other domains of mathematics, including complex analysis, partial differential equations and ergodic theory besides many more. The paper is an attempt to draw attention to certain applications of the Hahn-Banach theorem which are less familiar to the mathematical community, apart from highlighting certain aspects of the Hahn-Banach phenomena which have spurred intense research activity over the past few years, especially involving operator analogues and nonlinear variants of this theorem.For a discussion of a whole lot of issues related to the Hahn-Banach theorem not treated in this paper, the best source is a famous survey paper by Narici and Beckenstein [31] which deals, among other things, with the different settings witnessing the validity of the Hahn-Banach theorem.

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“…In this article we provide geometric conditions on a Banach space that imply the existence of a polynomial defined on a closed subspace which does not extend or, equivalently, the existence of a closed subspace such that its symmetric tensor product is not closed in the symmetric tensor product of the space. In this way, the extension of polynomials is studied as in [11], where the attention is focused on the geometry of the space. It is worth mentioning that, thus far, besides the spaces having type 2 (respectively, the spaces isomorphic to a Hilbert space), it is not known whether there exists a Banach space such that every quadratic form (resp.…”

Section: Introductionmentioning

confidence: 99%

“…It is worth mentioning that, thus far, besides the spaces having type 2 (respectively, the spaces isomorphic to a Hilbert space), it is not known whether there exists a Banach space such that every quadratic form (resp. every homogeneous polynomial of degree d > 2) defined on a subspace of it extends to the whole space (see Problems 1 and 2 in [11]).…”

Section: Introductionmentioning

confidence: 99%

“…The degree 2 case is studied in Section 2. We use the local nature of the property (see [11,Theorem 1.3]) to prove that if a Banach space E has no type p for some 1 < p < 2, then there exist a subspace and a quadratic form defined on it which does not extend (see Theorem 2.2). The case of polynomials of degree d > 2 is studied in Section 3.…”

Section: Introductionmentioning

confidence: 99%

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Existence of polynomials on subspaces without extension

Fernández-Unzueta

2013

Proc. Amer. Math. Soc.

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We prove the existence of a polynomial of degree d d defined on a closed subspace that cannot be extended to the Banach space E E (in particular, the existence of a nonextendible polynomial) in the following cases: (1) d ≥ 2 d\geq 2 and E E does not have type p p for some 1 > p > 2 1>p>2 ; (2) the space ℓ k \ell _k , k ∈ N k\in \mathbb {N} , 2 > k ≤ d 2>k\leq d , is finitely representable in E E . In each of these cases we prove, equivalently, the existence of a closed subspace F ⊂ E F\subset E such that the subspace ⊗ ^ s , π d F \hat {\otimes }^{d}_{s,\pi }{F} is not closed in ⊗ ^ s , π d E \hat {\otimes }^{d}_{s,\pi }{E} .

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Extension of polynomials defined on subspaces

Cited by 9 publications

References 21 publications

Banach limits: some new thoughts and perspectives

Banach limits: some new thoughts and perspectives

Some problems in functional analysis inspired by Hahn-Banach type theorems

Existence of polynomials on subspaces without extension

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