Closed graph theorem and continuity

Question

Let $V,W$ be banach spaces and $T:Vto W$ be linear map.
The closed graph theorem says $T$ is continuous iff the graph of $T$ is closed .
Graph of $T$ is closed is same as saying, if ( $x_nto x$ and $Tx_nto y$ ), Then $Tx=y$.
It's very similar to continuity and difference is here we say if $Tx_nto y$ (in other words, we can assume the convergence for granted) , but in continuity we have to show $Tx_n$ converges (and to $Tx$ ).
I want to look at the counter example where one doesn't implies other. I know in hausdroff space continuity implies closed graph. But I couldn't find an example in which closed graph doesn't implies continuity.  Please help.
Also if my understanding of closed graph theorem is wrong,then please correct me.

daw · Answer

In order to find a counter-example, one has to give up completeness.
Let $V=W=c_{00}$ be the space of real-valued sequences with at most finitely many non-zero entries. Supplied with $sup$-norm. This is a normed space but not complete.
Define $T$ by
$$
Tx = (x_1, 2x_2, dots, nx_n, dots),
$$
which is a linear mapping from $c_{00}$ to $c_{00}$. It is not continuous, since it is unbounded. However its graph is closed: $x_n to x$ and $Tx_n to y$ imply
$x_{n,k}to x_k$ for all $k$, as well as $kx_{n,k} to y_k$, hence $kx_k=y_k$ for all $k$, and $Tx=y$.
(We could have chosen $W=l^infty$ as well. Then the example still works.)

Closed graph theorem and continuity

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