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  "body": "\n## Summary\n\nA family of [[seminorms|FP5GVCN8GV]] on a vector space defines a topology.\nIn it, a net converges exactly when it converges for every seminorm of the family.\nWith this topology the vector space is a topological vector space, and in fact a [[locally convex space|B6GY2KJMWW]].\nThis is the standard way to put a topology on function spaces such as spaces of smooth functions, where no single norm is suitable.\n\n## Explanations\n\n### L1 human\n\nSometimes one way of measuring size is not enough, and one uses many measuring sticks at once.\nTwo vectors are declared close when they are close for finitely many of the sticks at a time.\nFor example, on the space of continuous functions on $\\R$, the seminorms $p_n(f) = \\sup_{\\abs{t} \\le n} \\abs{f(t)}$ give the topology of uniform convergence on bounded intervals.\nA sequence $f_k$ converges to $f$ in this topology when $p_n(f_k - f) \\to 0$ for each $n$ separately.\nThe functions $f_k(t) = t / k$ converge to $0$ in this sense, although they do not converge uniformly on all of $\\R$.\n",
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