Characterizing filters by convergence (with respect to filters) in Banach spaces

Let X be a topological space and let F be a filter on N , recall that a sequence ( x n ) n ∈ N in X is said to be F -convergent to the point x ∈ X , if for each neighborhood U of x, { n ∈ N : x n ∈ U } ∈ F . By using F -convergence in ℓ 1 and in Banach spaces, we characterize the P-filters, the P-filters+, the weak P-filters, the Q-filters, the Q-filters+, the weak Q-filters, the selective filters and the selective+ filters.