On a characterization of convergence in Banach spaces with a Schauder basis

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