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  "body": "\n## Definition\n\nLet $E$ be a real Hausdorff [[locally convex space|BW9BBF244W]].\nA sequence $(x_n)$ in $E$ is *Mackey–Cauchy* if there are an absolutely convex [[bounded set|3BAJ8ZTVPX]] $B \\subseteq E$ and real numbers $\\mu_{n,m}$ with this property.\nOne has $\\mu_{n,m} \\to 0$ as $n, m \\to \\infty$, and $x_n - x_m \\in \\mu_{n,m} B$ for all $n, m$.\nThe space $E$ is *convenient*, or *$c^\\infty$-complete*, if every Mackey–Cauchy sequence in $E$ converges in $E$.\n\n## Remarks\n\nKriegl and Michor give equivalent conditions.\nOne is that a curve $c \\colon \\R \\to E$ is smooth as soon as $\\ell \\circ c$ is smooth for every continuous linear functional $\\ell$.\nAnother is that the Riemann integral $\\int_0^1 c(t) \\, dt$ exists in $E$ for every smooth curve $c$.\n\n\nExample: every Fréchet space is convenient.\nA Mackey–Cauchy sequence is Cauchy, because $\\mu_{n,m} B$ is eventually inside any given neighbourhood of $0$ when $B$ is bounded, and a Fréchet space is complete.\nMore generally every sequentially complete locally convex space is convenient.\n\nNon-example: the space $P$ of real polynomials on $[0, 1]$ with the supremum norm is not convenient.\nIn a normed space, a sequence is Mackey–Cauchy if and only if it is Cauchy, since the unit ball is bounded.\nThe Taylor polynomials of $e^x$ form a Cauchy sequence in $P$ whose uniform limit $e^x$ is not a polynomial, so the sequence does not converge in $P$.\n\nA normed space is therefore convenient exactly when it is a [[Banach space|0BDTTM24DP]].\n",
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