For $A$ a ring, its analytic spectrum or Berkovich spectrum $Spec_an A$ is the set of all non-zero multiplicative seminorms on $A$, regarded as a topological space when equipped with the weakest topology such that all functions
of the form
for $a \in A$ are continuous.
If $A$ is equipped with the structure of a Banach ring, one takes the bounded multiplicative seminorms.
So a point in the analytic spectrum of $A$ corresponds to a non-zero function
to the real numbers, such that for all $x, y \in A$
${\Vert x \Vert} \geq 0$;
${\Vert x y \Vert} = {\Vert x \Vert} {\Vert y \Vert}$;
${\Vert x + y \Vert} \leq {\Vert x \Vert} + {\Vert y \Vert}$
and boundedness means that there exists $C \gt 0$ such that for all $x \in A$
For $k$ a field and $k[T]$ the polynomial ring over $k$ in one generator,
is the analytic affine line over $k$.
If $k = \mathbb{C}$, then $\mathbb{A}_k = \mathbb{C}$ is the ordinary complex plane.
The notion originates in
An exposition of examples of Berkovich spectra is in
Course notes are around definition 4 of
and section 1 of