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Do the dual spaces satisfy equivalence relation? For example the dual space of $\mathbb{R^n}$ is $\mathbb{R^n}$. Do the remaining two properties of equivalence relation can be applied to dual spaces, in general. that is,

$Reflexivity, Symmetry, Transitivity$.

Note: The dual space of $l^1$ is $l^\infty$. Can the dual space of $l^\infty$ be $l^1$?

I don't have any clue about the generalization of these properties. Any efforts will be appreciated, thanks in advance.

WKhan
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  • The dual space of $\Bbb R^n$ is not $\Bbb R^n$, it is the space of linear maps $\Bbb R^n\to \Bbb R$ and happens to be isomorphic to $\Bbb R^n$. – Hagen von Eitzen Apr 15 '18 at 10:14
  • more briefly it can be written as the dual space of $\mathbb{R^n}$ is $\mathbb{R^n}$ due to the homomorphism property. – WKhan Apr 15 '18 at 10:17
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    look at this https://en.wikipedia.org/wiki/Sequence_space – Kroki Apr 15 '18 at 10:18
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    $X$ dual of $Y$ does not imply $Y$ dual of $X$,$X$ dual of $Y$ and $Y$ dual of $Z$ does not imply $X$ dual of $Z$, and $X$ is its own dual (up to isometric isomorphism) in some special cases (e.g. Hilbert spaces). – Kavi Rama Murthy Apr 15 '18 at 12:00
  • For infinite-dimensional normed spaces this is false: the dual of $X$ is $Y$ implies the dual of $Y$ is $X$. So the term "dual" becomes misleading in that setting... Some people may therefore prefer to say "conjugate space" and not "dual space". – GEdgar Apr 15 '18 at 12:07
  • so the dual space of $l^\infty$ can be $l^1$? – WKhan Apr 15 '18 at 12:09

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