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  "body": "\n## Definition\n\nLet $K$ be a field that is complete for a [[non-Archimedean absolute value|1H5F1BP6XS]] $\\abs{\\cdot}$.\nLet $A$ be a commutative [[Banach algebra|9KYWN94QDN]] over $K$ with unit, with norm $\\norm{\\cdot}$.\nA *bounded multiplicative seminorm* on $A$ is a map $x \\colon A \\to [0, \\infty)$, written $f \\mapsto \\abs{f(x)}$, such that:\n\n1. $\\abs{1(x)} = 1$ and $\\abs{(f + g)(x)} \\le \\abs{f(x)} + \\abs{g(x)}$ for all $f, g \\in A$;\n2. $\\abs{(fg)(x)} = \\abs{f(x)} \\, \\abs{g(x)}$ for all $f, g \\in A$;\n3. $\\abs{f(x)} \\le \\norm{f}$ for all $f \\in A$.\n\nThe *Berkovich spectrum* $\\mathcal{M}(A)$ is the set of bounded multiplicative [[seminorms|FP5GVCN8GV]] on $A$.\nIt carries the weakest topology for which the map $x \\mapsto \\abs{f(x)}$ is continuous for every $f \\in A$.\nA $K$-affinoid algebra is a quotient of a [[Tate algebra|B1TEQ98JZD]] or of one of its versions with other radii.\nFor such $A$, $\\mathcal{M}(A)$ is a *$K$-affinoid space*.\nA *Berkovich analytic space* over $K$ is a space glued from $K$-affinoid spaces along affinoid domains.\n\n## Remarks\n\nBerkovich proved that $\\mathcal{M}(A)$ is a nonempty compact Hausdorff space when $A \\ne 0$.\n\nExample: let $A = K\\langle T \\rangle$ be the Tate algebra of power series $\\sum_n a_n T^n$ with $a_n \\to 0$, with the Gauss norm $\\norm{\\sum a_n T^n} = \\max_n \\abs{a_n}$.\nTake $a \\in K$ with $\\abs{a} \\le 1$ and a real number $0 \\le r \\le 1$.\nExpand $f \\in A$ as $f = \\sum_n c_n (T - a)^n$ and set $\\abs{f(\\zeta_{a,r})} = \\max_n \\abs{c_n} r^n$.\nThis is a point $\\zeta_{a,r}$ of $\\mathcal{M}(A)$, the point of the disc of centre $a$ and radius $r$.\nClause 3 holds because $\\abs{c_n} \\le \\norm{f}$ when $\\abs{a} \\le 1$, and clause 2 is a form of Gauss's lemma.\nFor $a = 0$ and $r = 1$ it is the Gauss norm itself.\nFor $r = 0$ it is $f \\mapsto \\abs{f(a)}$, evaluation at $a$, a classical point.\n\nNon-example: the supremum over the two-point set $\\set{0, 1}$, namely $f \\mapsto \\max(\\abs{f(0)}, \\abs{f(1)})$, satisfies clauses 1 and 3 but not clause 2.\nFor $f = T$ and $g = T - 1$ both values are $1$, while $fg$ vanishes at both points and has value $0$.\n\nLet $X = \\operatorname{Spec} B$ be an affine variety of finite type over $K$.\nIts analytification $X^{\\mathrm{an}}$ is the set of multiplicative seminorms on $B$ that extend $\\abs{\\cdot}$ on $K$, with the same kind of topology.\n",
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