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Let $X$ be an Banach space, and $X^*$ the space of linear functionals on $X$. The dual of $X^*$ is called the bidual, and if the bidual $X^{**}=X$, we say that $X$ is a reflexive space. It is well known that the $L^p$-spaces ($1<p<\infty$) are reflexive.

Now, let us define the tridual to be the dual of the bidual, $X^{***}$. Are there spaces $X$ such that $X^{***}=X$? What about "reflexivity" with respect to n-duals? Does this have any application?

Tomasz Kania
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    It is not true that $X^{}=X$ means $X$ is reflexive. Actually, we need $JX=X^{}$, where $J:X\to X^{}$ is the canonical embedding. It is possible to have nonreflexive spaces satisfy $X^{}=X$, as we see for instance with the James space.

    Anyway, I'm not sure whether there are examples of $n$th-dual isomorphic spaces which are neither 1st- nor 2nd-dual isomorphic. However, every Hilbert space satisfies $\mathcal{H}=\mathcal{H}^*$, and so will be $n$th-dual isomorphic for every $n$.

     – Ben W May 02 '14 at 11:11
  • What do you mean by $=$? As hatsoff points out the usually written $X=X^{}$ should be more precisely $JX=X^{}$ where $J$ is the canonical embedding. However, I do not know of a "canonical" embedding $X\hookrightarrow X^{***}$. – Jochen May 02 '14 at 14:03
  • It is classical and (using the characterization of reflexivity by weak compactness of the unit ball) not too difficult to show that $X^*$ refexive implies $X$ refexive. – Jochen May 02 '14 at 14:05
  • Related question: http://math.stackexchange.com/q/113198/8580 – Tom Cooney May 02 '14 at 16:11
  • Would you mind accepting my answer? Or yo require more clarification? – Tomasz Kania Sep 01 '18 at 09:42

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I am not sure what you mean by the equality $X^{***} = X$, if you mean that $X$ and $X^{***}$ are isometrically isomorphic then the answer is no, it does not imply reflexivity.

There is a classical example $J$ of a Banach space constructed by R. C. James (the James space) with the property that $J$ is isometrically isomorphic to $J^{**}$, yet $J$ is not reflexive:

R. C. James, A non-reflexive Banach space isometric with its second conjugate space, Proc. Nat. Acad. Sci. U. S. A. 37, (1951). 174–177.

Take $X = J\oplus_2 J^*$. Then $X$ is not reflexive as it contains a non-reflexive subspace $J$ but

$X^{***} \equiv J^{***} \oplus_2 J^{****}\equiv J^* \oplus_2 J\equiv J\oplus_2 J^* = X,$

where by $\equiv$ I mean the existence of an isometric isomorphism. Of course, this answers your question in negative for all $n$ as well.

Tomasz Kania
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