Verification of conclusions regarding duality maps

Question

I have two conclusions drawn from two results. I want to know how valid these two conclusions are. Firstly

Consider the duality mapping(set-valued) $J:X \rightrightarrows X^{*}$ defined: $J(u) := \{ f \in X^{*}: \langle f,u \rangle = \Vert u \Vert^{2} = \Vert f \Vert^{2} \}$. We have the following property of the duality mapping: If we take $X$ to be a separable Banach space then if $X^{*}$ is strictly convex then $J$ is single-valued.

Consider also the following corollary of the Hahn-Banach Theorem.

Corollary: If $X$ is normed space. For all $x_{o} \in X$ there exists $f_{o} \in X^{*}$ such that $f_{0}(x_{o}) = \Vert x_{o}\Vert_{X}^{2}$ and $\Vert f_{o} \Vert_{X^{*}} = \Vert x_{o} \Vert_{X}$.

Now for the two conclusion consider $X$ as a separable, reflexive Banach space. If $X_{m} \subset X$ are finite-dimensional subspaces of dimension $m$ then I have the following two conclusions:

1. From the Corollary(noting that the $f_{o}$ mentioned is unique if $X^{*}$ is strictly convex or if $X$ is a Hilbert space) and Riesz-representation Theorem we can see that when in a Hilbert space the duality mapping $J$ is a single-valued surjective isometry(therefore also a homeomorphism) and we can therefore identify $X$ and $X^{*}$.

2. We note that the duality mapping $J_{m}$ is a surjective isometry follows from the Corollary and the additional properties the mapping has when on a strictly convex space(in this case a Hilbert space.) $X_{m}$ and $X_{m}^{*}$ are finite-dimensional and are therefore not only isomorphic to each other but are also isomorphic to a Euclidean space of finite-dimension. We can therefore consider $X_{m}$ and $X_{m}^{*}$ as being equivalent to the Euclidean space of some finite-dimension with the usual norm. We note that the Euclidean space with the usual norm is a Hilbert space. In a Hilbert space the duality mapping is again a surjective isomtry. We can therefore identify $X_{m}$ and $X_{m}^{*}$, and we of course have $\Vert J_{m}x \Vert_{X^{*}_{m}} = \Vert x \Vert_{X_{m}}$.

Are these conclusions fine? Are there any recommendation? Are there any redundant statements or statements that are simply incorrect?

Thanks.

Answer 1

The obvious missing piece is: what do you mean by saying we can identify two spaces? The natural interpretation is that "we can think of them as being the same space". But that's not very precise. A more precise form is: we know a canonical isometric isomorphism between these space. This is how I understand the term, at least

To your first question. If $X$ is over real scalars, then $J$ is an isometric isomorphism between $X$ and $X^*$: it is a bijective linear map that preserves the norm. So I'll say yes, we can identify them. (But there are cases when we should not, even though we can.)

If $X$ is over complex scalars, then $J$ is not a linear map: it is conjugate linear. One could say that $X^*$ can be identified with $\overline{X}$, a version of $X$ in which the scalar multiplication is defined differently, as $\alpha *x = \bar\alpha x$.

To your second question: "being equivalent to the Euclidean space of some finite-dimension with the usual norm" introduces yet another undefined term, being equivalent. What does that mean?

Those spaces are isomorphic in the sense that there is a linear bijection that is continuous both ways. But they are not isometrically isomorphic in general: being isometrically isomorphic requires a linear bijection that preserves the norm. People do not usually consider two isomorphic normed spaces as being "the same". Doing so would throw the entire theory of the geometry of finite-dimensional normed spaces out of the window.