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It is also shown that this result is optimal in that when $p \\neq 2$ there are two computable copies of $\\ell^p$ with the property that any oracle that computes a linear isometry of one onto the other must also compute the halting set. Thus, $\\ell^p$ is $\\Delta_2^0$-categorical and is computably categorical if and only if $p = 2$. 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