0

If a space $X$ is isomorphic to a dual space $Y$, then is that mean $X$ is also a dual space?

$X$ is isomorphic to $Y$ means there exist a linear map $T : X \rightarrow Y$ such that both T and $T^{-1}$ are bounded.

bob
  • 3,079
Smita
  • 155
  • No, it is only isomorphic to. An arbitrary (normed linear) space *is* generally not a space of linear functionals, even if it is isomorphic to such a space. For instance, $\ell^\infty\cong(\ell^1)'.$ Would you say that $\ell^\infty$ *is* a dual space? It is only a space of sequences. – Anne Bauval Oct 03 '23 at 16:24
  • Or, perhaps the other way: When we say $X$ is a dual space, we mean $X$ is isomorphic to a dual space. Or maybe we mean $X$ is isometric to a dual space. – GEdgar Oct 03 '23 at 16:35
  • 2
    What sort of spaces are you talking about? You say "linear" and "bounded" in there, so I guess these are normed linear spaces? – GEdgar Oct 03 '23 at 16:35
  • 1
    @GEdgar's comment lets me wonder: are you asking "if a (normed linear) space is "isomorphic" (in your sense) to some dual space, is it also isometrically isomorphic to some (other) dual space?"? If I understand correctly OnurOktay's (now deleted) link, the answer is negative. – Anne Bauval Oct 03 '23 at 22:29
  • It depends what you mean by dual space. – blargoner Oct 04 '23 at 03:00

0 Answers0