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  "body": "\n## Example\n\nLet $J$ be the set of real sequences $x = (x_n)_{n \\ge 1}$ with $x_n \\to 0$ and $\\norm{x}_J \u003c \\infty$, where\n$$\\norm{x}_J = \\sup \\Big( \\sum_{i=1}^{k-1} (x_{p_i} - x_{p_{i+1}})^2 \\Big)^{1/2}.$$\nThe supremum runs over all $k \\ge 2$ and all integers $p_1 \u003c p_2 \u003c \\cdots \u003c p_k$.\nThen $\\norm{\\cdot}_J$ is a norm on $J$, and $J$ is a [[Banach space|06GZ956657]], the James space.\n\nCheck that $x_n = 1/\\sqrt{n}$ lies in $J$.\nFor $p_1 \u003c \\cdots \u003c p_k$, the differences $a_i = x_{p_i} - x_{p_{i+1}}$ are positive.\nThen $\\sum_i a_i^2 \\le (\\sum_i a_i)^2 = (x_{p_1} - x_{p_k})^2 \\le 1$.\nSo $\\norm{x}_J \\le 1$, although $x$ is not in $\\ell^2$, since $\\sum_n 1/n$ diverges.\n\n## Remarks\n\nJames built this space in 1951.\nThe canonical image of $J$ in its [[bidual|QKR3D65TTH]] $J^{**}$ has codimension $1$, so $J$ is not [[reflexive|NN8RWPC8WG]].\nYet $J$ is isomorphic to $J^{**}$.\nSo a Banach space can be isomorphic to its bidual without being reflexive.\nThe norm vanishes only at $0$ because $x_n \\to 0$: the difference $x_p - x_q$ with $q$ large is close to $x_p$.\n\n## Explanations\n\n### L1 human\n\nThe norm of James space measures the total squared jumps of a sequence along any chosen subsequence.\nA sequence that slides down to $0$ without wiggling has a small norm, even if it decreases slowly.\nThe space is famous as a counterexample: it is as big as its double dual, but not in the canonical way.\n",
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