= Continuous functions on the p-adic integers
{title2=$\mathcal C(\mathbb Z_p,\mathbb Q_p)$}
The <continuous functions> from the <p-adic integers> to the <P-adic numbers> form a <Banach space> over $\mathbb Q_p$ with <supremum norm> $\|f\|_\infty=\sup_{x\in\mathbb Z_p}|f(x)|_p$. Compactness of the domain bounds the norm, and <uniform convergence> in the complete codomain proves completeness. Translation and the <forward difference operator> $\Delta f(x)=f(x+1)-f(x)$ are bounded operators, with $\|\Delta f\|_\infty\leq\|f\|_\infty$.
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