Non-Archimedean Riesz-Frechet Representation

We derive a Riesz-Frechet representation for bounded linear functionals defined on the p-adic Hilbert spaces introduced by Kalisch [Ann. of Math. (2), 1947]. We also notice the surprising difference between the Archimedean case and the non-Archimedean case (exact non-Archimedean version of Riesz-Frechet representation fails).