Lineability

The Search for Linearity in Mathematics

; Luis Bernal-Gonzalez ; Daniel M. Pellegrino ; Juan B. Seoane Sepulveda

Renewed interest in vector spaces and linear algebras has spurred the search for large algebraic structures composed of mathematical objects with special properties. Bringing together research that was otherwise scattered throughout the literature, Lineability: The Search for Linearity in Mathematics collects the main results on the conditions for the existence of large algebraic substructures. Les mer
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Vår pris: 826,-

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Leveringstid: Sendes innen 21 dager

Om boka

Renewed interest in vector spaces and linear algebras has spurred the search for large algebraic structures composed of mathematical objects with special properties. Bringing together research that was otherwise scattered throughout the literature, Lineability: The Search for Linearity in Mathematics collects the main results on the conditions for the existence of large algebraic substructures. It investigates lineability issues in a variety of areas, including real and complex analysis.





After presenting basic concepts about the existence of linear structures, the book discusses lineability properties of families of functions defined on a subset of the real line as well as the lineability of special families of holomorphic (or analytic) functions defined on some domain of the complex plane. It next focuses on spaces of sequences and spaces of integrable functions before covering the phenomenon of universality from an algebraic point of view. The authors then describe the linear structure of the set of zeros of a polynomial defined on a real or complex Banach space and explore specialized topics, such as the lineability of various families of vectors. The book concludes with an account of general techniques for discovering lineability in its diverse degrees.

Fakta

Innholdsfortegnelse

Preliminary Notions and Tools
Cardinal numbers
Cardinal arithmetic
Basic concepts and results of abstract and linear algebra
Residual subsets
Lineability, spaceability, algebrability, and their variants





Real Analysis
What one needs to know
Weierstrass' monsters
Differentiable nowhere monotone functions
Nowhere analytic functions and annulling functions
Surjections, Darboux functions, and related properties
Other properties related to the lack of continuity
Continuous functions that attain their maximum at only one point
Peano maps and space-filling curves





Complex Analysis
What one needs to know
Nonextendable holomorphic functions: genericity
Vector spaces of nonextendable functions
Nonextendability in the unit disc
Tamed entire functions
Wild behavior near the boundary
Nowhere Gevrey differentiability





Sequence Spaces, Measure Theory, and Integration
What one needs to know
Lineability and spaceability in sequence spaces
Non-contractive maps and spaceability in sequence spaces
Lineability and spaceability in Lp[0, 1]
Spaceability in Lebesgue spaces
Lineability in sets of norm attaining operators in sequence spaces
Riemann and Lebesgue integrable functions and spaceability





Universality, Hypercyclicity, and Chaos
What one needs to know
Universal elements and hypercyclic vectors
Lineability and dense-lineability of families of hypercyclic vectors
Wild behavior near the boundary, universal series, and lineability
Hypercyclicity and spaceability
Algebras of hypercyclic vectors
Supercyclicity and lineability
Frequent hypercyclicity and lineability
Distributional chaos and lineability





Zeros of Polynomials in Banach Spaces
What one needs to know
Zeros of polynomials: the results





Miscellaneous
Series in classical Banach spaces
Dirichlet series
Non-convergent Fourier series
Norm-attaining functionals
Annulling functions and sequences with finitely many zeros
Sierpinski-Zygmund functions
Non-Lipschitz functions with bounded gradient
The Denjoy-Clarkson property





General Techniques
What one needs to know
The negative side
When lineability implies dense-lineability
General results about spaceability
An algebrability criterion
Additivity and cardinal invariants: a brief account





Bibliography


Index





Exercises, Notes, and Remarks appear at the end of each chapter.

Om forfatteren

Richard M. Aron is a professor of mathematics at Kent State University. He is editor-in-chief of the Journal of Mathematical Analysis and Applications. He is also on the editorial boards of Revista de la Real Academia de Ciencias Exactas,