A nonlinear version of the Riesz isomorphism

Question

The present question regards the proof of the following theorem which is found in Adams' Sobolev spaces, §2.30 - 2.33. Here $(\Omega, \mathcal{M}, \mu)$ denotes some arbitrary measure space and $L^p$ always stands for $L^p(\Omega)$.

Riesz Representation Theorem Let $1<p<\infty$ and let $p'=p/(p-1)$. For any $v \in L^{p'}$ denote $L_v$ to be the linear functional on $L^p$ defined by the following equation: $$\langle L_v, w\rangle =\int_{\Omega} vw\, d\mu.$$ Then the mapping $v\mapsto L_v$ is an isometric isomorphism of $L^{p'}$ onto $[L^p]^\star$.

The most interesting thing to prove is that this mapping $L$ is surjective, which as far as I know is usually done by means of the Radon-Nikodym's theorem of measure theory. On the contrary, Adams' book employs uniform convexity of $L^p$ space and two of the four Clarkson's inequalities, precisely: $$\tag{38} \forall u, v \in L^p, 2\le p <\infty,\quad \left\lVert \frac{u+v}{2}\right\rVert_p^{p'}+\left\lVert \frac{u-v}{2}\right\rVert_p^{p'}\ge\left( \frac{1}{2}\lVert u \rVert_p^p+\frac{1}{2}\lVert v\rVert_p^p\right)^{p'-1}, $$ $$\tag{40} \forall u, v \in L^p, 1< p \le 2,\quad \left\lVert \frac{u+v}{2}\right\rVert_p^{p}+\left\lVert \frac{u-v}{2}\right\rVert_p^{p}\ge \frac{1}{2}\lVert u \rVert_p^p+\frac{1}{2}\lVert v\rVert_p^p. $$ His proof follows those steps:

1. Because of uniform convexity, there exists a duality mapping $$\left[ F\in \left( L^p\right)^\star \right] \to \left[\text{the unique}\ w\in L^p\ \text{s.t.}\ \lVert w\rVert_p=\lVert F \rVert_\star\ \text{and}\ \langle F, w\rangle=\lVert F\rVert_\star^2\right].$$
2. The duality mapping has an explicitly known left inverse $$\left[ L_v\in (L^p)^\star\ \text{where}\ v=\frac{\lvert w\rvert^{p-1}\text{signum}(w)}{\left(\int \lvert w\rvert^p\,d\mu\right)^{\frac{p-2}{p}}}\right] \leftarrow \left[ w\in L^p\right].$$
3. Because of inequalities (38) and (40), the duality mapping is injective.

This means that the duality mapping is bijective and so, in particular, that any linear functional $T$ is of the form $L_v$, which is what Adams wanted to prove.

However, it seems to me that he actually proved much more than that: namely, this proof introduces the duality mapping, which is (as far as I can understand) a generalization of the Riesz isomorphism between a Hilbert space and its dual. Also, it looks like this mapping only depends on some easily generalizable properties of $L^p$ such as uniform convexity. So:

Question How much is it known about the duality mapping in an abstract Banach space? What are the minimal hypotheses that guarantee its existence? In what spaces has it got an explicit analytical expression?

Thank you for reading.

Answer 1

For every normed space $X$ you have duality with its dual via $$ \langle\cdot,\cdot\rangle: X\times X^*\to\mathbb{C}:(x,f)\mapsto f(x) $$ This duality is bilinear, and what is more $$ \Vert x\Vert=\sup\{|\langle x, f\rangle|: f\in X^*,\;\Vert f\Vert\leq 1\}\qquad \Vert f\Vert=\sup\{|\langle x, f\rangle|: x\in X,\;\Vert x\Vert\leq 1\} $$ Thus good duality always exist, the question is explicit description of $X^*$. The problem of complete description of duals of normed spaces seems to be completely hopeless. But for most common spaces there are some. Here are some of them

Let $(\Omega,\Sigma,\mu)$ be a measure space, then $$ L_p(\Omega,\Sigma,\mu)^*= \begin{cases} L_{p/(p-1)}(\Omega,\Sigma,\mu)\quad&\text{ if }\quad p\in(1,+\infty)\\ L_{\infty}(\Omega,\Sigma,\mu)\quad&\text{ if }\quad p=1\\ \mathrm{ba}(\Omega,\Sigma,\mu)\quad&\text{ if }\quad p=\infty \end{cases} $$ where $\mathrm{ba}(\Omega,\Sigma,\mu)$ is a space of finitely additive signed bounded measures on $(\Omega,\Sigma,\mu)$.

Let $\Omega$ be a normal space, then $$ C(\Omega)^*=\mathrm{rba}(\Omega) $$ where $rba(\Omega)$ is a space of regular bounded finitely-additive complex-valued measures on algebra generated by closed sets.

Let $\Omega$ be locally compact Hausdorff space, then $$ C(\Omega)^*=\mathrm{rca}(\Omega) $$ where $\mathrm{rca}(\Omega)$ is a space of regular bounded $\sigma$-additive complex-valued Borel measures on $\Omega$.

Proofs of these results you can find in Linear Operators, General Theory by N. Dunford, J. T. Schwartz

There is a non-commutative analogue for $L_p$ duality. Let $H$, $K$ be Hilbert spaces, then $$ S_p(H,K)^*= \begin{cases} S_{p/(p-1)}(K,H)\quad&\text{ if }\quad p\in(1,+\infty)\\ \mathcal{B}(K,H)\quad&\text{ if }\quad p=1\\ S_{\infty}(K,H)\quad&\text{ if }\quad p=\infty \end{cases} $$ Where $S_p(H,K)$ is $p$-th Shatten class operators.

Feel free to add some other dualities here.