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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?

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?

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?

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?

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?

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?