Application of Closed Graph Theorem

**Mimi89** · January 30th 2011, 06:41 AM

Hi all,

Any help (hints included) with this is appreciated!

Let $( \alpha_{n} )$ be a sequence of scalars such taht $\sum \alpha_{n} \beta_{n} $ converges whenever $( \beta_{n} ) \in c_{0} $ . Show that there is a linear operator $T: c_{0} \rightarrow l^{1} $ given by

$T(\beta_{n}) = (\alpha_{n} , \beta_{n}) $

Show that $T$ has a closed graph. Use the Closed Graph Theorem to deduce that $(\alpha_{n}) \in l^{1} $

**Drexel28** · January 31st 2011, 12:32 PM

Originally Posted by Mimi89

Hi all,

Any help (hints included) with this is appreciated!

Let $( \alpha_{n} )$ be a sequence of scalars such taht $\sum \alpha_{n} \beta_{n} $ converges whenever $( \beta_{n} ) \in c_{0} $ . Show that there is a linear operator $T: c_{0} \rightarrow l^{1} $ given by

$T(\beta_{n}) = (\alpha_{n} , \beta_{n}) $

Show that $T$ has a closed graph. Use the Closed Graph Theorem to deduce that $(\alpha_{n}) \in l^{1} $