# Lineability

## The Search for Linearity in Mathematics

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

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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.

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.

Luis Bernal Gonzalez 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 Paraiba. 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 Sao Paulo).

Juan B. Seoane Sepulveda 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 Cadiz jointly with the University of Karlsruhe and his second PhD from Kent State University.