Theorem 1 - Let $(X,\| \cdot \|)$ be a normed vector space. $X$ is a Banach space if and only if every absolutely convergent series in $X$ is convergent, i.e., whenever $\sum_n \|x_n\| < \infty,$ $\sum_n x_n$ converges in $X$ .

Theorem 2 - Let $X, Y$ be normed vector spaces, $X \ne 0$ . Let $B(X,Y)$ be the space of bounded operators $X \longrightarrow Y$ . Then $Y$ is a Banach space if and only if $B(X,Y)$ is a Banach space.