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  "body": "\n## Example\n\nLet $c_0$ be the set of sequences $x = (x_n)_{n \\ge 1}$ of real (or complex) numbers with $x_n \\to 0$, with the norm $\\norm{x}_\\infty = \\sup_n \\abs{x_n}$.\nThen $c_0$ is a [[sequence space|XYTN19EPF2]] and a [[Banach space|0BDTTM24DP]].\n\nCheck: $c_0$ is the kernel of the linear functional $x \\mapsto \\lim_n x_n$ on the [[space of convergent sequences|GWCBGYG5RA]] $c$.\nThis functional satisfies $\\abs{\\lim_n x_n} \\le \\norm{x}_\\infty$, so it is continuous, and its kernel is a closed linear subspace of $c$.\nA closed subspace of a Banach space is complete, and $c$ is a Banach space, so $c_0$ is a Banach space.\n\n## Remarks\n\nThe unit vectors $e_k$, with $1$ in place $k$ and $0$ elsewhere, form a [[Schauder basis|528G11SDWR]] of $c_0$: for $x \\in c_0$, $\\norm{x - \\sum_{k \\le N} x_k e_k}_\\infty = \\sup_{k \u003e N} \\abs{x_k} \\to 0$.\nIn particular $c_0$ is separable, unlike the space $\\ell^\\infty$ of [[bounded sequences|G3F2CWQSQH]].\nThe dual space of $c_0$ is isometrically isomorphic to $\\ell^1$, through the pairing $(a, x) \\mapsto \\sum_n a_n x_n$.\nThe space $c$ is the direct sum of $c_0$ and the line of constant sequences.\n\n## Explanations\n\n### L1 human\n\nThe null sequences are the sequences that tend to $0$, such as $1, 1/2, 1/3, \\dots$.\nThey form a vector space, and the largest absolute value of the terms is a good way to measure their size.\nThis space is one of the basic examples of an infinite-dimensional Banach space.\n",
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