Lineability

This book is a compendium of the currently known results on the size of linear and algebraic substructures within different classes of real or complex valued functions, which, as a whole, do not have such structures. The classes of such functions always contain some examples that serve as counterexamples in different mathematical settings. The work presented in this text facilitates better understanding of such examples. In general, this is a very well-written book that will be great reading for anybody interested in a true understanding of the riches of the structures of real and complex valued functions. -Krzysztof Chris Ciesielski, Professor of Mathematics, West Virginia University, and Adjunct Professor of Radiology, University of Pennsylvania, USA Hippasus of Metapontum shocked mathematicians of Pythagoras's school claiming the irrationality of the root of 2. Cantor astonished the mathematical community with the proof of the uncountability of irrational numbers. Weierstrass's monster of a continuous everywhere but differentiable nowhere frightened mathematicians with this counterintuitive example. Every single mathematician remembers the first time in her life when she discovers one of these three results. Gurariy proved in the sixties that the set of Weierstrass's monsters in the interval contains a vector space of countably infinite dimension. This result resonated in the last years and motivated the study of the existence of large algebraic structures of shocking mathematical objects, attracting the interest of many mathematicians from analysis, algebra, and topology. This excellent book provides the first systematic treatment of the quest for linearity in nonlinear topics. I encourage other mathematicians to take part in this adventure. They will be astonished after reading it! -J. Alberto Conejero, IUMPA, Universitat Politecnica de Valencia, Spain The late 2000s and early 2010s witnessed an explosion of papers on lineability/spaceability, which is the search for linearity on nonlinear problems. This trend in modern analysis investigates the existence-or not-of a (infinite dimensional, closed, dense) subspace of a given topological vector space formed, up to the null vector, by elements enjoying a certain prescribed distinguished property. Problems of completely different natures are studied, and here comes the relevance of this book: this is the first time all these different problems and solutions are assembled in book form. The book is very well written and everyone interested in the field should have it as permanent companion. -Geraldo Botelho, Universidade Federal de Uberlandia, Brazil Ever since the discovery of continuous and nowhere differentiable functions by Weierstrass, the study of exotic objects in analysis has fascinated mathematicians. While it has been known for some time that the sets of such exceptional objects are often topologically large, researchers have recently even uncovered large algebraic structures within these sets. This excellent book provides the first comprehensive treatment of the search for linearity in fundamentally nonlinear situations. It covers a wide variety of strange analytical objects and discusses their lineability properties. The few general techniques that are available are also presented. Each chapter contains a useful list of exercises and a notes-and-remarks section that directs the reader to the extensive literature in the field. The book is ideally suited for a one-semester course on lineability and related notions, and it is essential reading for anybody interested in the fascinating story of strange objects in analysis. -Karl Grosse-Erdmann, Department of Mathematics, Universite de Mons, Belgium Mathematical 'monsters' (functions satisfying certain pathologies) are, despite the conventional wisdom, more common and ubiquitous than they may appear at first glance. This wonderful book helps you find large vector spaces or linear algebras of such specimens. The book is a delicious piece of art that introduces the elegant linearity inside of nonlinear problems. -Domingo Garcia, University of Valencia, Spain