In mathematics, a Berkovich space, introduced by Berkovich (1990), is a version of an analytic space over a non-Archimedean field (e.g. p-adic field), refining Tate's notion of a rigid analytic space.

In the complex case, algebraic geometry begins by defining the complex affine space to be ${\displaystyle \mathbb {C} ^{n}.}$ For each ${\displaystyle U\subset \mathbb {C} ^{n},}$ we define ${\displaystyle {\mathcal {O}}_{U},}$ the ring of analytic functions on ${\displaystyle U}$ to be the ring of holomorphic functions, i.e. functions on ${\displaystyle U}$ that can be written as a convergent power series in a neighborhood of each point.

We then define a local model space for ${\displaystyle f_{1},\ldots ,f_{n}\in {\mathcal {O}}_{U}}$ to be

with ${\displaystyle {\mathcal {O}}_{X}={\mathcal {O}}_{U}/(f_{1},\ldots ,f_{n}).}$ A complex analytic space is a locally ringed ${\displaystyle \mathbb {C} }$-space ${\displaystyle (Y,{\mathcal {O}}_{Y})}$ which is locally isomorphic to a local model space.

When ${\displaystyle k}$ is a complete non-Archimedean field, we have that ${\displaystyle k}$ is totally disconnected. In such a case, if we continue with the same definition as in the complex case, we wouldn't get a good analytic theory. Berkovich gave a definition which gives nice analytic spaces over such ${\displaystyle k}$, and also gives back the usual definition over ${\displaystyle \mathbb {C} .}$

In addition to defining analytic functions over non-Archimedean fields, Berkovich spaces also have a nice underlying topological space.

A seminorm on a ring ${\displaystyle A}$ is a non-constant function ${\displaystyle |\!-\!|:A\to \mathbb {R} _{\geq 0}}$ such that

for all ${\displaystyle f,g\in A}$. It is called multiplicative if ${\displaystyle |fg|=|f||g|}$ and is called a norm if ${\displaystyle |f|=0}$ implies ${\displaystyle f=0}$.