Hello. I have some concern.
Question 1. In this version of the closed graph theorem, the mentioned operator is defined as $T:D(T)=X\to Y$?
Question 2. If I have have that an operator $T:D(T)\subset X\to Y$, does the theorem not apply to this operator?
Question 3. If $D(T)$ is dense in $X$, then is it applicable?
Question 4. If $T:D(T)\subset X\to X$ with $D(T)$ dense in $X$, then $T$ is bounded if and only if $T$ closed? For this question, what I have is the following:
$T$ closed $\Rightarrow (D(T),|\cdot |_{T})$ is a Banach space with $|\cdot|_{T}$ graph norm. Now, the operator $T:(D(T),|\cdot|_{T})\to (X,|\cdot|_{X})$ is closed, then by Closed graph theorem, $T:(D(T),|\cdot|_{T})\to (X,|\cdot|_{X})$ is bounded, i.e. $$|Tx|_{X}\leq C|x|_{T}\text{ for all }x\in D(T)$$ This is equivalent to $$|Tx|_{X}\leq C(|x|_{X}+|Tx|_{X}),\quad x\in D(T)$$ or $$|Tx|_{X}\leq \frac{C}{1-C}|x|_{X},\quad x\in D(T)$$ and from here, I would like to see if it is possible to extend the above inequality to all $x\in X$ using the density of $D(T)$ over $X$. Is it possible to do this? Also I don't know if the constant $\frac{C}{1-C}$ is non-negative.
These questions would help me a lot to understand what I am studying.
Thanks.