Continuous functions on the p-adic integers ($\mathcal{C}({\mathbb{Z}}_p,{\mathbb{Q}}_p)$)

The continuous functions from the p-adic integers to the P-adic numbers form a Banach space over ${\mathbb{Q}}_p$ with supremum norm $\Vert f{\Vert }_{\infty }={\sup }_{x\in {\mathbb{Z}}_p}|f(x){|}_p$. 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 $\Vert \Delta f{\Vert }_{\infty }\le \Vert f{\Vert }_{\infty }$.

Every function in continuous functions on the p-adic integers has a unique uniformly convergent expansion in the binomial polynomials, with $a_n\to 0$ and $a_n=({\Delta }^{n}f)(0)$. Conversely, any coefficient sequence tending to zero defines such a continuous function. To prove the expansion once $a_n\to 0$ is known, use $|(\genfrac{}{}{0ex}{}{x}{n}){|}_p\le 1$ for uniform convergence. Finite binomial inversion gives agreement with $f$ at each nonnegative integer, and density gives agreement everywhere. The coefficients recover successively from these integer values, proving uniqueness. Moreover $\Vert f{\Vert }_{\infty }={\sup }_n|a_n{|}_p$, because finite differences bound each coefficient by the norm and the expansion gives the reverse inequality.

The Mahler coefficients tend to zero, so the displayed series defines a continuous function and has value zero at zero. The Pascal's identity and uniform convergence give $\Delta Jf=f$. Thus the forward difference operator is surjective on continuous functions on the p-adic integers. Its kernel consists of constants: period one implies agreement on the dense nonnegative integers, and continuity then implies constancy. This selects the unique discrete antiderivative vanishing at zero.

There is also a direct construction on locally constant functions. If $h$ has period $M=p^r$, put $a_j=h(j)$, $S={\sum }_{i=0}^{M-1}{a}_i$, and $P_j={\sum }_{i<j}{a}_i$. For $x=j+My$ with $0\le j<M$ and $y\in {\mathbb{Z}}_p$, setThis is a continuous function on each residue class. Increasing $j$ gives $Jh(x+1)-Jh(x)=a_j$; at $j=M-1$, the next point has residue zero and quotient $y+1$, giving the same identity. The ultrametric inequality gives $\Vert Jh{\Vert }_{\infty }\le \Vert h{\Vert }_{\infty }$, and $Jh(0)=0$. The construction is independent of the chosen period, since two normalized discrete antiderivatives agree on the nonnegative integers and then on the p-adic integers by continuity. It is linear on the locally constant functions, which form a dense subset in the supremum norm. Completeness therefore extends it to every continuous function, retaining $\Delta J=I$ and the norm bound. This proves surjectivity without first using the Mahler theorem.

Let $T:\mathcal{C}({\mathbb{Z}}_p,{\mathbb{Q}}_p)\to {\mathbb{Q}}_p$ be linear and invariant under translation by one. For any $f$, choose its continuous discrete antiderivative $Jf$. Then $T(f)=T(\Delta Jf)=T(Jf(\cdot +1))-T(Jf)=0$. No continuity or boundedness of $T$ is assumed. In particular this applies to a form invariant under every translation by a p-adic integer.

For a continuous function $f:{\mathbb{Z}}_p\to {\mathbb{Q}}_p$, its nth Mahler coefficient isThis is the explicit iterate of the forward difference operator. These coefficients are exactly those of its Mahler expansion.

Approximate a continuous function on ${\mathbb{Z}}_p$ uniformly by a function constant on residue classes modulo $p^r$. On this finite space, translation $S$ satisfies $S^{p^r}=I$, so ${\Delta }^{p^r}=(S-I{)}^{p^r}$ has every matrix coefficient divisible by $p$. Hence $\Vert {\Delta }^{j{p}^r}\Vert \le p^{-j}$, and the Mahler coefficients of a locally constant function tend to zero. Since $|a_n(f)-a_n(g){|}_p\le \Vert f-g{\Vert }_{\infty }$, uniform approximation proves the assertion for every continuous function. This completes the coefficient-decay part of the Mahler theorem.