We know that $(\ell^2,\|.\|_2)$ is a reflexive space. Thus $(\ell^2,\|.\|)$ is reflexive for any norm $\|.\|$ on $\ell^2$ equivalent to $\|.\|_2$. Thus $(\ell^2,\|.\|)^{**}=(\ell^2,\|.\|)$. My question is what can we say about $(\ell^2,\|.\|)^{*}$? Is it $(\ell^2,\|.\|^{\prime})$ for some norm $\|.\|^{\prime}$ on $\ell^2$ or it can be some other space. Any suggestion is appreciated.
2026-02-23 02:35:33.1771814133
Reflexivity and dual space
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If you replace the norm in $\ell^2$ by an equivalent norm, you will not change the continuous linear functionals. Any linear functional which is bounded w.r.t. $\|\cdot\|_2$ will also be bounded w.r.t. $\|\cdot\|$ and vice versa. Hence, the dual space is the same but it is endowed with a different norm.