Topological and algebraic genericity and spaceability for an extended chain of sequence spaces

Abstract

Given a pair of topological vector spaces $$X,\; Y$$ where X is a proper linear subspace of Y it is examined whether $$Y\setminus X$$ is residual in Y (topological genericity), whether $$Y\setminus X$$ contains a dense linear subspace of Y except 0 (algebraic genericity) and whether $$Y\setminus X$$ contains a closed infinite dimensional subspace of Y except 0 (spaceability). In the present paper the spaces X and Y are either sequence spaces or spaces of analytic functions on the unit disc regarded as sequence spaces via the identification of a function with the sequence of its Taylor coefficients. For the spaces under consideration we give an affirmative answer to each of these questions providing general proofs which extend previous results.