This is a foundational theorem of p-adic analysis (Mahler; see Colmez). The coefficients are given explicitly by iterated finite differences: defining the discrete derivatives \phi^{[0]} = \phi and \phi^{[k+1]}(x) = \phi^{[k]}(x+1) - \phi^{[k]}(x), one has a_n(\phi) = \phi^{[n]}(0). The bound a_n(\phi) \to 0 and the isometry \vC(\phi) = \inf_n \vp(a_n(\phi)) express that the \binomc{x}{n} are an orthonormal basis; uniqueness follows because the basis is free over \ell^0_\infty(\N, L).

This is essentially a restatement of the orthonormal basis property. By Theorem 3.4.2, any \phi \in \cC(\Zp, \OL) is \phi = \sum_n a_n(\phi)\binomc{x}{n} with a_n(\phi) \to 0, so by continuity and linearity a measure \mu is determined by the numbers c_n = \int_{\Zp} \binomc{x}{n} \cdot \mu through \int_{\Zp} \phi \cdot \mu = \sum_n a_n(\phi) c_n. Conversely any g = \sum_n c_n T^n \in \OL[[T]] defines, by exactly this formula, a measure \mu_g with \Am_{\mu_g} = g; the sum converges in \OL because a_n(\phi) \to 0 and c_n \in \OL. Thus \Am is a bijection. For multiplicativity it suffices, by density, to check on Dirac measures Definition 3.3.4: since 1 topologically generates (\Zp, +) and 1+T topologically generates \OL[[T]], the assignment [1] \mapsto 1+T identifies \Lam(\Zp) with \OL[[T]], and \delta_a \mapsto (1+T)^a matches the top arrow \delta_a \mapsto \Am_{\delta_a} = (1+T)^a. Inverting p gives \sM(\Zp, L) \cong \OL[[T]][1/p].