Suppose $X$ is a normed linear space. If for every Banach space $Y$ and for every linear operator $T:X\to Y$, graph of $T$ is closed implies $T$ is continuous, then can we prove that $X$ is a Banach space?



Converse of closed graph theorem
No. The closed graph theorem in this form is equivalent to $X$ being a barreled space. See item 15 [here.](https://en.wikipedia.org/wiki/Barrelled_space#Characterizations_of_barreled_spaces)


There are incomplete normed spaces that are barreled. See [here.](https://link.springer.com/article/10.1007/BF03322742)


