# Thread: Application of Closed Graph Theorem

1. ## Application of Closed Graph Theorem

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}
$

2. 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}
$
This question doesn't make msuch sense. What is $c_0$? Is it $\ell^\infty$?

3. $c_0$ is the space of all sequences converging to 0. It is a subspace of $l^\infty$.