Lineability The Search for Linearity in Mathematics

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.

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 vectorsWild behavior near the boundary, universal series, and lineabilityHypercyclicity 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.

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.