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  "body": "\n## Definition\n\nLet $\\mathbb{K}$ be $\\R$ or $\\C$, and let $X$ be a [[sequence space|XYTN19EPF2]] over $\\mathbb{K}$.\nAn *FK-space* is such an $X$ together with a topology $\\tau$ on $X$ such that:\n\n1. $(X, \\tau)$ is a [[Fréchet space|3GAA614673]], that is, a complete metrizable locally convex topological vector space.\n2. For each $n$, the coordinate map $X \\to \\mathbb{K}$, $x \\mapsto x_n$, is [[continuous|0X3WGX95FE]].\n\n## Remarks\n\nExample: the space $\\omega$ of all sequences, with the [[product topology|0GBG8B49AA]] of $\\mathbb{K}^{\\N}$, is an FK-space.\nThis topology is given by the seminorms $p_n(x) = \\abs{x_n}$, so it is locally convex and makes each coordinate continuous.\nIt is metrizable by $d(x, y) = \\sum_n 2^{-n} \\min(1, \\abs{x_n - y_n})$, and it is complete because a Cauchy sequence converges in each coordinate.\n\nExample: each [[space ℓ^p|08P6T2WH6R]] with $1 \\le p \\le \\infty$ and its usual norm is an FK-space, since $\\abs{x_n} \\le \\norm{x}_p$.\n\nNon-example: the space of sequences with finitely many nonzero terms, with the norm $\\sup_n \\abs{x_n}$, is not an FK-space under this norm, because it is not complete.\nThe sequences $(1, 1/2, \\dots, 1/k, 0, 0, \\dots)$ form a Cauchy sequence whose coordinatewise limit $(1/n)$ is not in the space.\n\nAn FK-space whose topology is given by a complete norm is a BK-space.\nBy the closed graph theorem, a sequence space has at most one FK topology, and every inclusion map between FK-spaces is continuous.\n",
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