instituto de matemáticas universidad de sevilla
Antonio de Castro Brzezicki
On a problem by Vladimir Gurariy concerning subspaces of continuous functions
Let  A ⸦ R and denote by C^(A) the subset of C(A) of functions attaining
their maximum at a unique point. In 2004 V. I. Gurariy and L. Quarta
proved that the set C^([0, 1]) U {0} does not contain a 2-dimensional space whereas C^([0, 1]) U {0} and  C^(R) U {0}  do contain a 2-dimensional space. Using the usual teminology in lineability theory, we can say that C^([0, 1]) is not 2-lineable whereas C^(R) and C^([0,1)) are 2-lineable. During a Non-linear Analysis Seminar held at Kent State University in the academic year 2003/2004, V. I. Gurariy posed the following question:
                 Is C^(R) (or equivalently C^([0 1))) n-lineable for n ≥ 3?
The answer to this question has resisted the efforts of many mathematicians ever since. Using a topological approach based on Moore's Theorem we have been able to prove (among other results) that C^(R) is not 3-lineable. Some generalizations will also be presented.
Joint work with L. Bernal-González, G. A. Muñoz-Fernández and J.B. Seoane-