p-adic Banach spaces & families of modular forms

This paper is dedicated to Bernard Dwork who has been a friend and an inspiration for many years. Let p be a prime, Cp the completion of an algebraic closure of the p-adicnumbers Qp and K a finite extension of Qp contained in Cp. Let v be the valuation on Cp such that v(p) = 1 and let | | be the absolute value on Cp such that |x | = p−v(x) for x ∈ Cp