Last week, our functional analysis course covered Riesz Representation theorem for $L^p(X,\mu),(1\leq p < \infty)$, namely, $(L^p(X,\mu))^* = L^q(X,\mu)$. And I was stuck with this homework problem to work out the dual space of a more general version of $L^p$ space:
Let $X$ be a Banach space and assume both $X$ and $X^*$ are known, define $L^p([0,1],X)=\{f(t):[0,1]\rightarrow X\big{|}\int_0^1 ||f(t)||^p_X\text{d}t<\infty\}$ where $||\cdot||_X $ is norm in $X$ and define similar $L^p$ norm:$||f||_p = \big{(}\int_0^1 ||f(t)||^p_X\text{d}t\big{)}^\frac{1}{p}$. The question is to determine what is [$L^p([0,1],X)]^*$?
My first guess is $[L^p([0,1],X)]^*=L^q([0,1],X^*)$, where $\frac{1}{p}+\frac{1}{q}=1$. To begin with, if we are given $g(s)\in L^q([0,1],X^*)$ and define $T_g\in [L^p([0,1],X)]^* $ by $$ T_g(f) = \int_0^1 g(s)(f(s)) \text{d}s $$ Apparently, since each $g(s)$ is a linear functional of $X$,then $T(g)$ is a linear functional. Besides, \begin{align} |T_g(f)|\leq\int_{0}^{1} |g(s)(f(s))| \mathrm{d} s &\leq \int_{0}^{1} ||g(s)||_{X^*}\cdot||f(s)||_X \mathrm{d} s\\ \leq (\int_{0}^{1}\|g(s)\|_{X^*}^{q} \mathrm{d} s)^{\frac{1}{q}}\cdot(\int_{0}^{1}\|f(s)\|_{X}^{p} \mathrm{d} s)^{\frac{1}{p}} = &||g||_{q}\cdot ||f||_p \end{align} I use Hölder's inequality above. This justifies $T(g)\in $[$L^p([0,1],X)]^*$ and $||T_g||\leq ||g||_q$. However, I got stuck to preceed to verify next two things:
Find $f \in L^{q}([0,1],X)$ such that $T_g(f) \geq ||g||_q$
Given $T\in \left[L^{p}([0,1], X)\right]^{*}$, how to find $g\in L^{q}\left([0,1], X^{*}\right)$ such that $T(f)=\int_{0}^{1} g(s)(f(s)) \mathrm{d} s$ holds for any $f\in L^p([0,1],X)$?
I try to invoke Radon- Nikodym theorem by first defining characteristic funtion $\chi_E(t) = \begin{cases} x_0 &t\in E\\ 0, & t \notin E\end{cases}$, where $x_0$ is a unit vector in $X$ and $E \subset [0,1]$, so we have $\chi_E \in L^{p}([0,1], X)$. Next, we define $\nu(E) = T(\chi_E)$, then we prove $\nu$ is indeed a signed measure and $\nu<< m$, where $m$ denotes Lebesgue measure here. By Radon-Nikodym theorem, there exists $g:[0,1]\rightarrow \mathbb{R}$ such that $\nu(E) = \int_0^1 ||\chi_E(s)||_X\cdot g(t) \text{d}s $ but I hope to get $g:[0,1]\rightarrow X^*$ such that $\nu(E) = \int_0^1 g(s)(\chi_E(s)) \text{d}s $
One reason why the definition $T_g(f)=\int_{0}^{1} g(s)(f(s)) \mathrm{d} s$ might be reasonable is because I can let $X=\mathbb{R}$, then $T_g(f)=\int_{0}^{1} g(s)(f(s)) \mathrm{d} s = \int_{0}^{1} g(s)\cdot f(s) \mathrm{d} s$ as the normal $L^p$ space.
Sorry about the length of this question. I try to write out what I can get so far. If you have any insights, please share with me! Thanks in advance.