 In contrast to the classical theory of "p"-adic analytic manifolds, rigid analytic spaces admit meaningful notions of analytic continuation and connectedness.The basic rigid analytic object is the "n"-dimensional unit polydisc, whose ring of functions is the Tate algebra formula_1, made of power series in "n" variables whose coefficients approach zero in some complete nonarchimedean field "k".The Tate algebra is the completion of the polynomial ring in "n" variables under the Gauss norm (taking the supremum of coefficients), and the polydisc plays a role analogous to that of affine "n"-space in algebraic geometry.