Lineability and algebrability of pathological phenomena in analysis

doi:10.1016/j.jmaa.2006.03.025

Journal of Mathematical Analysis and Applications

Volume 326, Issue 2, 15 February 2007, Pages 929-939

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Abstract

We show that, in analysis, many pathological phenomena occur more often than one could expect, that is, in a linear or algebraic way. We show this by means of the construction of large algebraic structures (infinite dimensional vector spaces or infinitely generated algebras) enjoying some special or pathological properties.

Keywords

Lineability

Algebrability

Spaceability

Hamel basis

Taylor expansion

Linear functions

R

-linear functions

Dirichlet function

Everywhere surjective functions

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¹: The research of the author was partially supported by the Academy of Finland Project No. 205644.