Lineability: The Search for Linearity in Mathematics

Lineability: The Search for Linearity in Mathematics

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Product Details

ISBN-13: 9781482299090
Publisher: Taylor & Francis
Publication date: 10/14/2015
Series: Chapman & Hall/CRC Monographs and Research Notes in Mathematics Series , #14
Pages: 328
Product dimensions: 6.10(w) x 9.40(h) x 0.90(d)

About the Author

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, Fisicas y Naturales. Serie A. Matematicas (RACSAM) and the Mathematical Proceedings of the Royal Irish Academy. His primary research interests include functional and nonlinear analysis. He received his PhD from the University of Rochester.

Luis Bernal González is a full professor at the University of Seville. His main research interests are complex analysis, operator theory, and the interdisciplinary subject of lineability. He is the author or coauthor of more than 80 papers in these areas, many of them concerning the structure of the sets of mathematical objects. He is also a reviewer for several journals. He received his PhD in mathematics from the University of Seville.

Daniel M. Pellegrino is an associate professor at the Federal University of Paraíba. He is also a researcher at the National Council for Scientific and Technological Development (CNPq) in Brazil. He is an elected affiliate member of the Brazilian Academy of Sciences and a young fellow of The World Academy of Sciences (TWAS). He received his PhD in mathematical analysis from Unicamp (State University of São Paulo).

Juan B. Seoane Sepúlveda is a professor at the Complutense University of Madrid. He is the coauthor of over 100 papers. His main research interests include real and complex analysis, operator theory, number theory, geometry of Banach spaces, and lineability. He received his first PhD from the University of Cádiz jointly with the University of Karlsruhe and his second PhD from Kent State University.

Table of Contents

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
Sierpiński-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.

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