  Given a densely defined and closed operator $A$ acting on a complex Hilbert
space $\mathcal{H}$, we establish a one-to-one correspondence between its
closed extensions and subspaces $\mathfrak{M}\subset\mathcal{D}(A^*)$, that are
closed with respect to the graph norm of $A^*$ and satisfy certain conditions.
In particular, this will allow us to characterize all densely defined and
closed restrictions of $A^*$. After this, we will express our results using the
language of Gel'fand triples generalizing the well-known results for the
selfadjoint case.
  As applications we construct: (i) a sequence of densely defined operators
that converge in the generalized sense to a non-densely defined operator, (ii)
a non-closable extension of a symmetric operator and (iii) selfadjoint
extensions of Laplacians with a generalized boundary condition.
