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  "body": "\n## Definition\n\nLet $\\mathbb{K}$ be $\\R$ or $\\C$, let $V$ be a [[vector space|05EGHEQZAW]] over $\\mathbb{K}$, and let $\\mathcal{P}$ be a family of [[seminorms|FP5GVCN8GV]] on $V$.\nFor $x \\in V$, finitely many $p_1, \\ldots, p_n \\in \\mathcal{P}$ and $\\varepsilon \u003e 0$, put\n$$B(x; p_1, \\ldots, p_n; \\varepsilon) = \\set{y \\in V : p_i(y - x) \u003c \\varepsilon \\text{ for } i = 1, \\ldots, n}.$$\nThe *topology generated by* $\\mathcal{P}$ is the [[topology|0CQHFPGJ45]] on $V$ defined as follows.\nA set $U \\subseteq V$ is open if for every $x \\in U$ there are $p_1, \\ldots, p_n \\in \\mathcal{P}$ and $\\varepsilon \u003e 0$ with $B(x; p_1, \\ldots, p_n; \\varepsilon) \\subseteq U$.\n\n## Remarks\n\nWith this topology $V$ is a topological vector space, each $p \\in \\mathcal{P}$ is continuous, and it is the coarsest topology on $V$ with these two properties.\nA net $(x_i)$ converges to $x$ if and only if $p(x_i - x) \\to 0$ for every $p \\in \\mathcal{P}$.\nThe topology is Hausdorff if and only if $\\mathcal{P}$ separates points: for each $x \\ne 0$ some $p \\in \\mathcal{P}$ has $p(x) \\ne 0$.\nExample: on $\\R^X$, the space of all real functions on a set $X$, the seminorms $p_x(f) = \\abs{f(x)}$ for $x \\in X$ generate the topology of pointwise convergence, which is the product topology.\nThe check: a basic set $B(f; p_{x_1}, \\ldots, p_{x_n}; \\varepsilon)$ restricts the values at $x_1, \\ldots, x_n$ only, as a basic open set of the product topology does.\nNon-example: for $0 \u003c p \u003c 1$, the usual metric topology of $L^p([0, 1])$ is not generated by any family of seminorms.\nIndeed, its only continuous linear form is $0$.\nA nonzero continuous seminorm would give a nonzero continuous linear form by the Hahn–Banach theorem.\nThe topologies generated by families of seminorms are exactly the topologies of [[locally convex spaces|B6GY2KJMWW]].\n",
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