The authors investigate topological and algebraic properties of the following chain of sequence spaces \[ A^\infty(\mathbb D) \subsetneq \bigcap_{p>0} \ell_p \subsetneq \ell_p \subsetneq c_0 \subsetneq \ell_\infty \subsetneq H(\mathbb D) \subsetneq {\mathbb C}^{\mathbb{N}_0}, \] where \(H(\mathbb D)\) is the space of all holomorphic functions on the open unit disc \(\mathbb D\) of the complex space \(\mathbb C\), and \(A^\infty(\mathbb D)\) is the space of holomorphic functions \(f\) on \(\mathbb D\) such that all derivatives \(f^{(i)}\) can be continuously extended on \(\overline{\mathbb{D}}\). Both spaces, \(H(\mathbb D)\) and \(A^\infty(\mathbb D)\), are seen as sequence spaces via the identification of a holomorphic function with the sequence of its Taylor coefficients.
Given a topological vector space \(Y\) and a proper linear subspace \(X\) of \(Y\), it is said that the pair \((X,Y)\) has: topological genericity if \(Y \setminus X\) is residual in \(Y\), or equivalently if \(X\) is contained in an \(F_\sigma\)-meager subset of \(Y\); algebraic genericity if there exists a linear subspace dense in \(Y\) and also contained in \((Y \setminus X) \cup \{0\}\); and spaceability if \((Y \setminus X) \cup \{0\}\) contains a closed infinite-dimensional subspace of \(Y\). The main results of the paper provide that any pair \((X,Y)\) of the chain of sequence spaces mentioned above has the properties of topological and algebraic genericity and spaceability.