In the present paper, the authors study topological genericity, algebraic genericity and spaceability of \(\ell^p\) spaces and of intersections of them, as factors in a chain \(X_i \subset X_j, X_i \neq X_j\) for \(i < j\). The aforementioned notions are defined via properties of \(X_j \setminus X_i\) with respect to the space \(X_j\). Namely, in the chain context, these notions are relative to a pair of members of the chain. So, one speaks about topological genericity whenever, \(X_j \setminus X_i\) is a \(G_\delta\)-dense subset of the space \(X_j\) or about algebraic genericity in case, except for \(0\), \(X_j \setminus X_i\) contains a vector space, dense in \(X_j\), while spaceability is gained whenever \(X_j \setminus X_i\) contains except for \(0\), an infinite-dimensional vector space, closed in \(X_j\).
Details of some of the proofs that concern the above three properties for \(\ell^p\) spaces, are included in [the second author, “A project about chains of spaces regarding topological and algebraic genericity and spaceability”, Preprint (2020), arXiv:2005.01023], see also [Bull. Am. Math. Soc. 51, No. 1, 71–130 (2014; Zbl 1292.46004)] by the first author et al. Further, the authors study topological or algebraic genericity, and spaceability for other chains with certain types of spaces, or for chains of holomorphic functions, as, e.g., Hardy spaces \(H^p\) on the unit disc. One of the main results, concerning the algebraic genericity of the latter spaces, assures that for \(0<p<q<+\infty\), the set \((H^p \setminus H^q)\cup \{0\}\) contains a vector space which is dense in \(H^p\). The presentation of the results is clear and it is enriched by detailed comments on the form of the used spaces in the referred chains.