Such spaces were introduced by John Tate in 1962, as an outgrowth of his work on uniformizing p-adic elliptic curves with bad reduction using the multiplicative group.
The basic rigid analytic object is the n-dimensional unit polydisc, whose ring of functions is the Tate algebra
, 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.
describing a locally ringed G-topologized space with a sheaf of k-algebras, such that there is a covering by open subspaces isomorphic to affinoids.
Huber worked out a theory of adic spaces to resolve this, by taking a limit over all blow-ups.
Vladimir Berkovich reformulated much of the theory of rigid analytic spaces in the late 1980s, using a generalization of the notion of Gelfand spectrum for commutative unital C*-algebras.
Since the topology is pulled back from the real line, Berkovich spectra have many nice properties, such as compactness, path-connectedness, and metrizability.