ABSTRACT
The study of lineability and spaceability (although not with these terms) within the context of measure spaces can be traced back to, at least, the 1950’s. In fact, from a famous theorem due to Grothendieck (see [352, Chapter 6]) we can conclude that if 0 < p <∞ and µ is finite, then L∞(X,µ) is not spaceable in Lp(X,µ). Also, if one looks at Rosenthal’s work (e.g., [348,349]) we essentially find the proof of the spaceability of `∞r c0. This chapter deals with lineability and spaceability results in the lines of the latter ones, in the framework of Measure Theory and Sequence Spaces. Since the results presented here also encompass `p spaces for 0 < p < 1 and, in this case, the `p spaces are not Banach spaces, we present some basic facts from the theory of quasi-Banach spaces (which are not easy to find in Functional Analysis textbooks).
