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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?

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