Let $X,||\cdot||_X$ be a Banach Space, $A$ a dense linear subspace of $B$ and $T\colon A\to X$ a linear map ($T$ not necessarily bounded). Then $||\cdot||_A:=||\cdot||_X+||T\cdot||_X$ makes $A$ into a normed linear space. If we define $\tilde{A}$ to be the abstract closure of $A$ with respect to this norm, then we can naturally consider $\tilde{A}$ as a subspace of $X$, as $X$ is complete, so $$ A\subset \tilde{A}\subset X. $$ We can then define $\tilde{T}\colon \tilde{A}\to X$ to be the unique extension of $f$ from $A$ to $\tilde{A}$.
Then $(\tilde{A},\tilde{T})$ is a closed densely defined operator in the sense of Theorem 7.4.2 of Evans PDE 2nd Edition.
This seems to imply that any densely defined linear operator can be extended to a unique maximal closed (in the sense of densely defined operators) one. Is this correct?
The only references I could find dealt with Hilbert spaces and extending symmetric operators, in which case uniqueness of a self-adjoint extension did not hold.