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  "body": "\n## Definition\n\nLet $\\mathbb{K}$ be $\\R$ or $\\C$, and let $X$ be a [[sequence space|XYTN19EPF2]] over $\\mathbb{K}$.\nA *BK-space* is such an $X$ together with a norm $\\norm{\\cdot}$ on $X$ such that:\n\n1. $(X, \\norm{\\cdot})$ is a [[Banach space|0BDTTM24DP]].\n2. For each $n$, the coordinate map $X \\to \\mathbb{K}$, $x \\mapsto x_n$, is [[continuous|0X3WGX95FE]].\n\n## Remarks\n\nEquivalently, a BK-space is an [[FK-space|71DMQHW6JZ]] whose topology is given by a norm.\n\nExample: $\\ell^\\infty$ with $\\norm{x}_\\infty = \\sup_n \\abs{x_n}$ is a BK-space.\nIt is a Banach space, and $\\abs{x_n} \\le \\norm{x}_\\infty$ shows that each coordinate map is continuous.\nIn the same way, the [[spaces ℓ^p|08P6T2WH6R]] for $1 \\le p \u003c \\infty$, the space $c$ of convergent sequences and the space $c_0$ of null sequences are BK-spaces.\n\nNon-example: the space $\\omega$ of all sequences is an FK-space but not a BK-space.\nIts FK topology, the topology of coordinatewise convergence, is not given by any norm, because no neighbourhood of $0$ in it is bounded.\n",
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