$\begingroup$

My question is how to prove part (c) of Problem 31 of Chapter 3 of Vol. 1 of Reed and Simon, "Functional Analysis".

Part (c) reads,

(c) Let $E^{**}$ be a separable Banach space and let $1\leq p<\infty$. Prove that $L^p(M, du, E)^*$ is naturally isometrically isomorphic to $L^q(M, du, E^*)$. (Hint: First show that it is enough to prove that every bounded linear transformation $T$ of $E$ into $L^q(M, d\mu)$ is of the form $[T(x)](m)=[f(m)](x)$ for some $f\in L^q(M, d\mu, E^*)$. Prove this in the special case where $E=l_1$. Finally use Problems 29 and 30 to treat the general separable Banach space, E.)

Note: while the text doesn't say this it is implied that $1/q + 1/p =1$.

Problem 29 guides the reader to the proof that any separable Banach space is isometrically isomorphic to a quotient of $l_1$.

Problem 30 reads,

Let $X$ be a Banach space and let $Y$ be a closed subspace of $X$. Let $Y^o$ in $X^*$ be defined by $Y^o=\left\{l\in X^*: l|_Y=0\right\}$. Given a bounded linear functional $f$ on $X/Y$, define $\pi^*(f)\in X^*$ by $[\pi^*(f)](x)=f([x])$. Prove that $\pi^*$ is an isometric isomorphism of $(X/Y)^*$ onto $Y^o$.

I can prove Problems 29 and 30 and I see how the application of these problems to Problem 31. (c) should proceed.

I can't prove the hint, however.

Specific questions:

I can see that $\lVert T\rVert \leq \lVert f\rVert$, so that every $f\in L^q(M, d\mu, E^*)$ induces $T\in \mathcal{L}(E,L^q(M,d\mu))$. I can't see why every such $T$ is induced this way. It is not at all clear to me how to prove that elements of $L^p(M,d\mu, E)^*$ should correspond to elements of $\mathcal{L}(E,L^q(M,d\mu))$. Given $f\in L^q(M,d\mu, E^*)$, define $F:L^p(M,d\mu,E)\to\mathbb{R}$ by $$F(g)=\int_M [f(m)](g(m))d\mu.$$ Since $$\lvert F(g)\rvert\leq \int_M\lvert [f(m)](g(m))\rvert d\mu\leq\int_M\lVert f(m)\rVert\lVert g(m)\rVert d\mu\leq \lVert f\rVert\lVert g\rVert.$$ This can be used to show that $L^q(M,d\mu,E^*)\subset L^p(M,d\mu,E)^*$. Is there a direct proof that this map $f\mapsto F$ is onto? I assume that there must be and it'd follow the normal proof that $L^p(M,d\mu)^*\approx L^q(M,d\mu)$ but I can't see my way through the detail.

For the record the full text of Problem 31 of Chapter 3 of Vol. 1 of Reed and Simon, "Functional Analysis" is,

(a) Let $E$ be a Banach space with separable dual and $<M, \mu>$ a measure space with $L^p(M, d\mu)$ separable for all $1<o<\infty$. Develop the theory of $L^p(M, d\mu, E)$ analogous to the theory of $L^2(M, d\mu, \mathcal{H})$ discussed in Sections II.1 and II.4.

(b) Prove that $L^p(M\times N, d\mu\otimes d

u)$ and $L^p(M,d\mu, L^p(N,d

u))$ are naturally isomorphic.

(c) Let $E^{**}$ be a separable Banach space and let $1\leq p<\infty$. Prove that $L^p(M, du, E)^*$ is naturally isometrically isomorphic to $L^q(M, du, E^*)$. (Hint: First show that it is enough to prove that every bounded linear transformation $T$ of $E$ into $L^q(M, d\mu)$ is of the form $[T(x)](m)=[f(m)](x)$ for some $f\in L^q(M, d\mu, E^*)$. Prove this in the special case where $E=l_1$. Finally use Problems 29 and 30 to treat the general separable Banach space, E.)