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My question was prompted by an earlier MO by @Daniel:

    Duality map in strictly convex Banach spaces

I will even use his symbol   $\phi$   below.

Let   $B$   be an arbitrary Banach space. Let   $S := \{x\in B:\|x\|=1\}$   be its unit sphere. Let   $\Gamma := \{f\in B^*: \|f\|=1\}$   be the unit sphere in the dual space $B^*$.

QUESTION   Are the following two conditions on $B$ equivalent:

  1. $B$   is isometric to a Hilbert space.
  2. There exists an isometry   $\phi: \Gamma \rightarrow S$   such that   $\forall_{f\in\Gamma}\ f(\phi(f))=1$.

?

The finite-dimensional case is especially basic.

REMARK 0   Perhaps similar questions were asked in the past (on MO too?)--please, let me know.

REMARK 2 The case of   $\mathbb R^2$   and its two dual but isometric norms   $L_\infty\quad L_1$   is interesting. The general question related to the one above is to describe all Banach spaces which are isometric to their dual space. Is there any beside the Hilbert spaces and   $\mathbb R^2$   with the norm(s) just mentioned above?

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2  
Adding a link to the earlier question (specially if you use the notation introduced there) would be useful. –  Mariano Suárez-Alvarez May 10 '13 at 22:47
    
I have tried to provide a link. The standard HTML way "a href=" didn't work, the link was ignored by the system (didn't show up). I'll learn the OM way one day, I promise myself. –  Włodzimierz Holsztyński May 11 '13 at 3:32
    
I am not using any essential prior non-standard notation. I was just giving credit to Daniel by mentioning his usage of $\phi$. I could use equally well most any letter. –  Włodzimierz Holsztyński May 11 '13 at 3:35
2  
Maybe this example should be mentionned here: let $X$ be reflexive (so for any finite dimensional space), then $X \oplus_2 X^*$ and its dual are isometric. –  Yanqi QIU May 11 '13 at 7:15
1  
Dear Wlodzimierz, you can find formatting tips on this page htpp://mathoverflow.net/editing-help; in particular, a common way to get a link you have to use the syntax [some text](the actual URL). I hope your time on MO is not that bad, because I for one enjoy very much reading your contributions! –  Mariano Suárez-Alvarez May 11 '13 at 23:48

2 Answers 2

up vote 9 down vote accepted

Yes it is true. Let me show that the existence of $\phi$ implies that the norm of $B^*$ is associated to an inner product. Then it follows easily that the spaces are Hilbert.

It suffices to verify that the norm is an inner product one on every two-dimensional subspace. (Indeed, this property is equivalent to the parallelogram law, which involves only two-dimensional configurations.)

Let $V$ be a two-dimensional subspace of $B^*$ (equipped with the restriction of the norm of $B^*$) and $V^*$ the dual of $V$ (equipped with the norm dual to this restriction). There is a natural map $\pi:B\to V^*$ dual to the inclusion $V\to B^*$. Namely $\pi(x)(f)=f(x)$ for $x\in B$, $f\in V$. Note that $\pi$ does not increase the norm: $\|\pi(x)\|\le\|x\|$ for all $x\in B$.

Let $D$ be the unit ball of $V$ and $E\subset V$ the maximum-area ellipse contained in $D$ (i.e., the John ellipsoid of $D$). Let $\Sigma$ be the set of points where the boundaries of $D$ and $E$ meet. It is easy to see that $\Sigma$ contains at least two pairs of opposite points. The ellipse $E$ is a unit ball of a Euclidean norm $\|\cdot\|_E$ on $V$. Since $E\subset D$, we have $\|f\|_E\ge\|f\|$ for all $f\in V$ and equality is attained only if $x$ is proportional to an element of $\Sigma$.

On $V^*$, there is a dual Euclidean norm, denoted by $\|\cdot\|^*_E$. There we have $\|y\|^*_E\le \|y\|$ for all $y\in V^*$. The norm $\|\cdot\|_E$ is associated to an inner product, which defines an isomorphism $I:V\to V^*$ preserving the Euclidean norm.

For every $f\in\Sigma$, we have $\pi(\phi(f))=I(f)$. Indeed, $\|\pi(\phi(f))\|\le\|\phi(f)\|=1$, hence $\|\pi(\phi(f))\|^*_E\le 1$. On the other hand, $\pi(\phi(f))(f)=f(\phi(f))=1$. Since $\|f\|_E=1$, this is possible only for $\pi(\phi(f))=I(f)$.

Now consider two linearly independent vectors $f,g\in\Sigma$ and look at the distance between $f$ and $-g$. Since $\phi$ is an isometry, we have $$ \|f+g\|=\|\phi(f)-\phi(-g)\| \ge \|\pi(\phi(f)-\phi(-g))\| . $$ The r.h.s equals $$ \|\pi(\phi(f))-\pi(\phi(-g))\| = \|I(f)-I(-g)\|=\|I(f+g)\| $$ since $\pi$ and $I$ are linear and $f,-g\in\Sigma$.

Thus $\|I(f+g)\|\le \|f+g\|$ and therefore $$ \|I(f+g)\|_E^*\le\|I(f+g)\|\le \|f+g\| \le \|f+g\|_E . $$ But $I$ preserves the Euclidean norm, so the inequalities turn into equalities. In particular, $\|f+g\|= \|f+g\|_E$, hence $f+g$ is proportional to an element of $\Sigma$.

Thus we have proved that, for every $f,g\in\Sigma$, the normalized bisector $\frac{f+g}{\|f+g\|}$ also belongs to $\Sigma$. Since $\Sigma$ is a closed subset of an ellipse and contains more than two points, it follows that $\Sigma$ is the whole ellipse. This means that $D=E$, so the norm on $V$ is Euclidean. Q.E.D.

Remark. The proof would be much easier (in fact, nearly trivial) if we assumed in advance that $\phi$ is a restriction of an isometry between $B^*$ and $B$. Then it would be linear by Mazur-Ulam and one could just define the inner product of $f,g\in B^*$ by $2\langle f,g\rangle=g(\phi(f))+f(\phi(g))$.

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Nice, Sergei. If some duality map is biLipschitz, then must the Banach space be isomorphic to a Hilbert space? I have a (possibly false) recollection that this is true but don't see a proof. –  Bill Johnson May 13 '13 at 7:43
    
@Bill: this is plausible but I don't know for sure. In finite dimensions, you can differentiate the map at some point and get that a codimension 1 section is within a bounded distance from a Euclidean norm. I don't know how to carry this over to infinite dimensions. –  Sergei Ivanov May 13 '13 at 9:54
    
@Sergei, somehow I missed your answer, I discovered your answer just a moment ago, I am really sorry. I'll carefully read your answer within 24h, and will comment on it (even if I will not get it :-), but at least I will admit it). –  Włodzimierz Holsztyński May 15 '13 at 7:13
    
@Sergei, your theorem/proof is very-very nice, you had an idea after an idea. –  Włodzimierz Holsztyński May 15 '13 at 23:39

A related question was asked earlier by Mark Meckes: Self-dual finite-dimensional complex normed spaces. He pointed out the $l^1$-$l^\infty$ example and noted that it generalizes to unit balls that are regular polygons in the real two-dimensional case. He also told us the $X \oplus_2 X^*$ construction.

I believe the specific question asked by Wlodzimierz has a positive answer, based on a comment I heard Giles Pisier make many years ago --- he said something very similar to this, though I don't remember exactly what. I don't have a reference though.

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