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  "body": "\n## Definition\n\nLet $V$ be a vector space over $\\mathbb{R}$ or $\\mathbb{C}$ with a [[norm|0J3578562S]] $\\|\\cdot\\|$, and let $d(x, y) = \\|x - y\\|$.\nThe space $(V, \\|\\cdot\\|)$ is a Banach space when the metric space $(V, d)$ is [[complete|08EWT7J0SC]].\n\n## Remarks\n\nExamples are $\\mathbb{R}^n$ with any norm, the spaces [[ell p|08P6T2WH6R]] for $1 \\le p \\le \\infty$, and $C_b(X)$ with the [[sup norm|0PARCKE8WP]].\nThe space of polynomial functions on $[0, 1]$ with the sup norm is not a Banach space. Its completion is $C([0, 1])$.\nIn a normed space $V$, completeness is the same as: [[each absolutely convergent series converges|082MGNWNA8]].\nEach [[finite-dimensional normed space is complete|0ABP2F2ZP7]].\n\n## Explanations\n\n### L1 human\n\nA Banach space is a vector space where you can measure lengths, and where Cauchy sequences always converge.\nSo you can take limits in it freely, as in $\\mathbb{R}^n$.\n",
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