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  "body": "\n## Example\n\nLet $p \\ge 1$ be a real number. Let $\\ell^p$ be the set of real sequences $x = (x_n)_{n \\ge 1}$ with $\\sum_{n \\ge 1} |x_n|^p \u003c \\infty$, and let\n$$\\|x\\|_p = \\left( \\sum_{n \\ge 1} |x_n|^p \\right)^{1/p}.$$\nLet $\\ell^\\infty$ be the set of bounded real sequences, with $\\|x\\|_\\infty = \\sup_{n \\ge 1} |x_n|$.\nThen each $\\ell^p$, for $1 \\le p \\le \\infty$, is a vector space, $\\|\\cdot\\|_p$ is a norm on it, and $\\ell^p$ is a [[Banach space|0BDTTM24DP]].\n\n## Remarks\n\nThe norms extend the [[p-norms on R^n|0KNJWEVSMG]] to infinite sequences.\nFor $1 \\le p \u003c q \\le \\infty$, we have $\\ell^p \\subseteq \\ell^q$, and the inclusion is strict. For example, $(1/n)$ is in $\\ell^2$ but not in $\\ell^1$.\nThe space $\\ell^2$ has the inner product $\\langle x, y \\rangle = \\sum_n x_n y_n$, and it is a Hilbert space.\nThe closed unit ball of $\\ell^p$ is not compact: the unit vectors $e_k$ satisfy $\\|e_j - e_k\\|_p = 2^{1/p}$ for $j \\neq k$ and $p \u003c \\infty$.\n\n## Explanations\n\n### L1 human\n\nThe spaces $\\ell^p$ are infinite-dimensional versions of $\\mathbb{R}^n$ with the p-norm.\nA sequence belongs to $\\ell^p$ when its terms go to $0$ fast enough for the sum of $|x_n|^p$ to be finite.\n",
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