  We extend the well-known characterizations of convergence in the spaces $l_p$
($1\le p<\infty$) 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.
