Consider the space $\ell_p=\{x=\{x_n\}: \sum|x_n|^p<\infty\}$ for $1\leq p<\infty$.We know that $\ell_p$ can be given $\|.\|_q$ for $q>p$ which is not equivalent to $\|.\|_p$,the usual norm.So, $\ell_p$ is not finite dimensional.Now can we construct a Hamel basis for $\ell_p$?I wonder if it is easy to construct a concrete basis for this Banach space.I also wonder if it is countable or uncountable.I also have similar question for $\ell_\infty=\{\text{All bounded sequences in }\mathbb C\}$.In that case also,is it easy to find a Hamel basis?
2026-03-26 09:41:11.1774518071
Hamel basis of $\ell_p$ and $\ell_\infty$.
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