On weak filter convergence and the radon-riesz type theorem

The author shows a characterization of a (unbounded) weakly filter convergent sequence which is parallel to that every weakly null sequence (xn) in a Banach space admits a norm null sequence (yn) with yn∈co(xk)k≥n for all n∈ℕ. A version of the Radon-Riesz type theorem is also proved within the frame of the filter convergence.