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  "body": "\n## Definition\n\nLet $\\mathbb{K}$ be $\\R$ or $\\C$.\nLet $\\omega$ be the set of all [[sequences|0QE8STGJH2]] $x = (x_n)_{n \\ge 1}$ with terms in $\\mathbb{K}$.\nWith termwise addition $(x + y)_n = x_n + y_n$ and scalar multiplication $(\\lambda x)_n = \\lambda x_n$, the set $\\omega$ is a [[vector space|05EGHEQZAW]] over $\\mathbb{K}$.\nA *sequence space* is a [[linear subspace|0EPTJA5QV4]] of $\\omega$.\n\n## Remarks\n\nExample: the set $\\ell^\\infty$ of bounded sequences is a sequence space.\nIt contains the zero sequence, and if $\\abs{x_n} \\le M$ and $\\abs{y_n} \\le N$ for all $n$, then $\\abs{x_n + y_n} \\le M + N$ and $\\abs{\\lambda x_n} \\le \\abs{\\lambda} M$.\nOther examples are $\\omega$ itself, the [[spaces ℓ^p|08P6T2WH6R]], the space $c$ of convergent sequences and the space $c_0$ of null sequences.\nAnother is the space $\\varphi$ of sequences with only finitely many nonzero terms.\n\nNon-example: the set of sequences with $x_1 = 1$ is not a sequence space, because it does not contain the zero sequence.\n\nThe word is often used for a sequence space together with a norm or a topology, as in the theory of FK-spaces and BK-spaces.\n",
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