Proposition 20 — the domain of an unbounded closed operator is not complete

Let $H$ and $Y$ be complex Hilbert spaces and let $X:D(X)\subseteq H\to Y$ be a linear operator defined on a subspace $D(X)$ of $H$. Assume that $X$ is closed (its graph is closed in $H\times Y$) and unbounded: there is no constant $C$ with $\|Xx\|\le C\|x\|$ for all $x\in D(X)$. Then the inner product space $D(X)$, with the norm of $H$, is not complete.

Completeness of the state space is incompatible with having an unbounded observable defined on every state.

Formalization Note The paper does not state a hypothesis on $X$ beyond unboundedness, but its proof appeals to the converse of the Hellinger–Toeplitz theorem, and without a hypothesis of this kind the statement is false (a discontinuous linear functional defined on all of $H$ has a complete domain). Closedness is the standard hypothesis covering self-adjoint operators such as position and momentum.