Let $V,W$ be banach spaces and $T:V\to W$ 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 ( $x_n\to x$ and $Tx_n\to y$ ), Then $Tx=y$.
It's very similar to continuity and difference is here we say if $Tx_n\to y$ (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.
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.)