Lineability and spaceability for the weak form of Peano’s theorem and vector-valued sequence spaces

Barroso, Cleon S.; Botelho, Geraldo; Fávaro, Vinícius V.; Pellegrino, Daniel

doi:10.1090/s0002-9939-2012-11466-2

Cited by 12 publications

(9 citation statements)

References 17 publications

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“…Many interesting results in the field have been discovered lately, mainly concerning lineability/spaceability in functions spaces [1,3,4,7,8,[18][19][20][21][22]27], sequence spaces [1,2,6,9,23], spaces of linear operators/homogeneous polynomials [10,11,23,28,31], and even spaces of holomorphic functions [24]. Nevertheless, maximal spaceability is rarely obtained.…”

Section: Introductionmentioning

confidence: 99%

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Maximal spaceability in sequence spaces

Botelho

Cariello

Fávaro

et al. 2012

Linear Algebra and its Applications

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Section: Introductionmentioning

confidence: 99%

“…In [6] it is proved that, for an infinite-dimensional Banach space X, Several results about Lorentz sequence spaces can be stated. We give just one example.…”

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confidence: 99%

Maximal spaceability in sequence spaces

Botelho

Cariello

Fávaro

et al. 2012

Linear Algebra and its Applications

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“…Before finishing this section, let us point out that (very recently) some results related to the problems treated in this section have been obtained by Barroso, Botelho, Fávaro, and Pellegrino in [36], and by G lab, Kaufmann, and Pellegrini [164,165]. Within the framework of Orlicz spaces, we also refer the reader to the very recent work by Akbarbaglu and Maghsoudi [4].…”

Section: Measurability and Integrationmentioning

confidence: 91%

Linear subsets of nonlinear sets in topological vector spaces

Bernal–González

Pellegrino

Seoane‐Sepúlveda

2013

Bull. Amer. Math. Soc.

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197

149

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For the last decade there has been a generalized trend in mathematics on the search for large algebraic structures (linear spaces, closed subspaces, or infinitely generated algebras) composed of mathematical objects enjoying certain special properties. This trend has caught the eye of many researchers and has also had a remarkable influence in real and complex analysis, operator theory, summability theory, polynomials in Banach spaces, hypercyclicity and chaos, and general functional analysis. This expository paper is devoted to providing an account on the advances and on the state of the art of this trend, nowadays known as lineability and spaceability. 109 5. Some remarks and conclusions. General techniques 113 About the authors 117 Acknowledgments 118 References 118Theorem 1.2 (Gurariy, 1966 [178]). The set of continuous nowhere differentiable functions on [0, 1] is lineable.Let us also recall that, although the set of everywhere differentiable functions in R is, in itself, an infinite dimensional vector space, in 1966 Gurariy obtained the following analogue of Theorem 1.1.License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use LINEAR SUBSETS OF NONLINEAR SETS IN TOPOLOGICAL VECTOR SPACES 73 Theorem 1.3 (Gurariy, 1966 [178]). The set of everywhere differentiable functions on [0, 1] is not spaceable in C[0, 1]. Also, there exist closed infinite dimensional subspaces of C[0, 1] all of whose members are differentiable on (0, 1).Somehow, what we are seeing is that what one could expect to be an isolated phenomenon can actually even have a nice algebraic structure (in the form of infinite dimensional subspaces). Unfortunately, and as we mentioned above, the Baire category theorem cannot be employed in the search for large subspaces such as the ones mentioned in the previous results. Let us now provide a more formal and complete definition for the above concepts and some other ones. Definition 1.4 (Lineability and spaceability [22,181,253]). Let X be a topological vector space and M a subset of X. Let μ be a cardinal number.(1) M is said to be μ-lineable (μ-spaceable) if M ∪ {0} contains a vector space (resp. a closed vector space) of dimension μ. At times, we shall refer to the set M as simply lineable or spaceable if the existing subspace is infinite dimensional.(2) We also let λ(M ) be the maximum cardinality (if it exists) of such a vector space. 1 (3) When the above linear space can be chosen to be dense in X, we shall say that M is μ-dense-lineable.Moreover, Bernal introduced in [69] the notion of maximal-lineable (and those of maximal-dense-lineable and maximal-spaceable), meaning that, when keeping the above notation, the dimension of the existing linear space equals dim(X). Besides asking for linear spaces, one could also study other structures, such as algebrability and some related ones, which were presented in [25,27,253]. Definition 1.5. Given a Banach algebra A, a subset B ⊂ A, and two cardinal numbers α and β, we say that:

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“…In this direction, we obtain the following result. The source of inspiration for this result is an idea present, for instance, in the recent papers [2,3,6], which suggest the usefulness in studying lineability and spaceability issues. In particular, Theorem 1.5 generalizes [6, Theorem 2.1].…”

Section: Introductionmentioning

confidence: 88%

An interplay between the weak form of Peano's theorem and structural aspects of Banach spaces

2013

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In this paper we establish some new results concerning the Cauchy-Peano problem in Banach spaces. Firstly, we prove that if a Banach space E admits a fundamental biorthogonal system, then there exists a continuous vector field f : E → E such that the autonomous differential equation u ′ = f (u) has no solutions at any time. The proof relies on a key result asserting that every infinite-dimensional Fréchet space with a fundamental biorthogonal system possesses a nontrivial separable quotient. The later, is the byproduct of a mixture of known results on barrelledness and two fundamental results of Banach space theory (namely, a result of Pe lczyński on Banach spaces containing L 1 (µ) and the ℓ 1 -theorem of Rosenthal). Next, we introduce a natural notion of weak-approximate solutions for the non-autonomous Cauchy-Peano problem in Banach spaces, and prove that a necessary and sufficient condition for the existence of such an approximation is the absence of ℓ 1 -isomorphs inside the underline space. We also study a kind of algebraic genericity for the Cauchy-Peano problem in spaces E having complemented subspaces with unconditional Schauder basis. It is proved that if K(E) denotes the family of all continuous vector fields f : E → E for which u ′ = f (u) has no solutions at any time, then K(E) {0} is spaceable in sense that it contains a closed infinite dimensional subspace of C(E), the locally convex space of all continuous vector fields on E with the linear topology of uniform convergence on bounded sets.MSC 2010: 34G20, 34A34, 46B20, 47H10, 34K07.

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Lineability and spaceability for the weak form of Peano’s theorem and vector-valued sequence spaces

Cited by 12 publications

References 17 publications

Maximal spaceability in sequence spaces

Maximal spaceability in sequence spaces

Linear subsets of nonlinear sets in topological vector spaces

An interplay between the weak form of Peano's theorem and structural aspects of Banach spaces

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