Is the range of this operator closed? I think I am stuck with showing closedness of the range of a given operator. Given a sequence $(X_n)$ of closed subspaces of a Banach space $X$. Define $Y=(\oplus_n X_n)_{\ell_2}$ and set $T\colon Y\to X$ by $T(x_n)_{n=1}^\infty = \sum_{n=1}^\infty \frac{x_n}{n}$. Is the range of $T$ closed?

The range is not necessarily closed. For example, if $X=(\oplus_{n \in \mathbb{N}} X_n)_{\ell_2}$=Y:
if $T(Y)$ is closed, $T(Y)$ is a Banach space. $T$ is a continuous bijective map from $X$ onto $T(Y)$, so is an homeomorphism (open map theorem) . But $T^{-1}$ is not continuous, because $T^{-1}(x_n)=nx_n$, for $x_n \in X_n$.
So $T(Y)$ is not closed.
