Convergence on a dense subspace under a uniform operator bound (${\sup }_h\Vert A_h\Vert <\infty ,A_{h}d\text{ converges on dense }D⟹A_{h}x\text{ converges for every }x$)

Let $A_h:X\to Z$, $0<h<1$, be bounded linear operators from a normed space into a Banach space, with $M={\sup }_h\Vert A_h\Vert <\infty$. If $A_{h}d$ converges as $h\to 0^{+}$ for each element of a dense subspace $D$, then $A_{h}x$ converges for every $x\in X$. The limit defines a bounded linear operator $A$ with $\Vert A\Vert \le M$.

For $d\in D$, the triangle inequality givesApproximate $x$ by $d$ and use the Cauchy criterion for a function limit on $D$. The completeness of $Z$ supplies the limit; passing to the limit preserves linearity and the bound. When every limit on $D$ is zero, the inequality $\Vert A_{h}x\Vert \le M\Vert x-d\Vert +\Vert A_{h}d\Vert$ shows $A=0$.