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?
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What do you mean by "construct"? Also, I have never seen an explicit (non-finite) Hamel basis in my life. Did you search for one? – JonathanZ Nov 19 '22 at 15:52
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@JonathanZsupportsMonicaC I have seen some, but not for complete normed space of infinite dimension. – Ryszard Szwarc Nov 19 '22 at 16:14
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@RyszardSzwarc - I guess there's always compactly supported sequences, and you can always take an arbitrary set and say "consider the vector space for which this is a Hamel basis", making one by fiat. I'm guessing the basic answer to this question is "No", but I lack the breadth of experience to say so. – JonathanZ Nov 19 '22 at 16:22
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2Does this answer your question? A Hamel basis for $\ell^p$? – Chris Eagle Nov 19 '22 at 20:51