On Banach space norms for I -bounded sequences and their quotient

In this note we establish that, for an an admissible ideal I, the I-lim sup can be used to give a norm which generates a Banach space topology for the quotient of the I bounded sequences modulo the I-null sequences via an elementary proof. Also the I-bounded and I-null sequences cannot, in general, be given a Banach or locally convex Frechet topology that admits continuous coordinate functionals.