On a Characterization of Convergence in Banach Spaces with a Schauder Basis

We extend the well-known characterizations of convergence in spaces $l_p$ ($1\le p<\infty$) $p$-summable sequence and $c_0$ vanishing sequences to a general characterization Banach space with Schauder basis obtain as instant corollaries an infinite-dimensional separable Hilbert $c$ convergent sequences.