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  "body": "\n## Definition\n\nLet $\\mathbb{K}$ be $\\R$ or $\\C$.\nA *Fréchet space* over $\\mathbb{K}$ is a topological vector space $E$ over $\\mathbb{K}$ that is both [[locally convex|B6GY2KJMWW]] and an [[F-space|86BYEV9AX8]].\n\n## Remarks\n\nEquivalently, the topology of $E$ is generated by a countable family of [[seminorms|FP5GVCN8GV]] $p_1 \\le p_2 \\le \\cdots$ that separates points, and $E$ is complete.\nFrom such seminorms one gets a translation-invariant metric\n$$d(x, y) = \\sum_{n = 1}^{\\infty} 2^{-n} \\frac{p_n(x - y)}{1 + p_n(x - y)}$$\nthat induces the topology.\nExample: the space $C(\\R)$ of continuous real functions with the seminorms $p_n(f) = \\sup_{\\abs{t} \\le n} \\abs{f(t)}$ is a Fréchet space.\nA sequence that is Cauchy for all $p_n$ converges uniformly on each $[-n, n]$, and the limit is continuous.\nEvery Banach space is a Fréchet space.\nNon-example: $L^p([0, 1])$ with $0 \u003c p \u003c 1$ is an F-space but is not locally convex, so it is not a Fréchet space.\nSome older texts use the name Fréchet space for what is here called an F-space.\n",
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