= Banach-Steinhaus convergence theorem
{c}
= Convergence on a dense set under a uniform operator bound
{synonym}
Let $X$ be a <normed space>, $Y$ a <Banach space>, and $A_n:X\to Y$ be <bounded linear operators> with $C=\sup_n\|A_n\|<\infty$. Convergence of $A_nd$ for every $d$ in a <dense subset> $D$ implies convergence of $A_nx$ for every $x\in X$ to a <bounded linear operator> $A$, with $\|A\|\leq\liminf_n\|A_n\|$. Indeed, $\|A_nx-A_mx\|\leq2C\|x-d\|+\|A_nd-A_md\|$ makes $(A_nx)$ a <Cauchy sequence>; completeness gives its limit, and linearity and the bound pass to that limit. If $X$ is a <Banach space>, pointwise convergence conversely implies the uniform bound by the <Uniform boundedness principle>. If the limit <bounded linear operator> is specified in advance, completeness of $Y$ is unnecessary.
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