Closable operator

Remarks

The following are equivalent.

1. The operator $T$ is closable.
2. The closure of the graph $\Gamma(T)$ in $X \times Y$ is the graph of an operator.
3. For every sequence $(x_n)$ in $D(T)$ with $x_n \to 0$ and $T x_n \to y$, one has $y = 0$.

Example: on $C[0, 1]$ with the supremum norm, let $D(T)$ be the space of polynomials and $Tp = p'$. If $p_n \to 0$ and $p_n' \to g$ uniformly, then $p_n(x) - p_n(0) = \int_0^x p_n'$ gives $0 = \int_0^x g$ for all $x$, so $g = 0$; hence $T$ is closable.

Non-example: on $L^2(0, 1)$, let $D(T) = C[0, 1]$ and $Tf = f(0) \cdot \mathbf{1}$, where $\mathbf{1}$ is the constant function $1$. The functions $f_n(x) = \max(0, 1 - n x)$ satisfy $f_n \to 0$ in $L^2(0, 1)$ but $T f_n = \mathbf{1}$ for all $n$, so $T$ is not closable.