The dual space of a product space

Question

Suppose $\prod^n L^1(I) $ is the product space of n $L^1$ integrable functions. What would be its dual space of continous linear functionals? would it be $\prod^n L^{\infty}(I)$? Do I need the norm for the product space to define the dual?

Answer 1

Let $X$ be any Banach space, $n \in \mathbb{N}$ and $1 \le p \le \infty$.

On the space $\prod_{i=1}^n X$ we define the norm \begin{equation*} \lVert (x_1, \ldots, x_n) \rVert := \Big( \sum_{i = 1}^n \lVert x_i \rVert_X^p \Big)^{1/p}. \end{equation*} In case $p = \infty$ this is to be understood as \begin{equation*} \lVert (x_1, \ldots, x_n) \rVert := \max_{i = 1,\ldots,n} \lVert x_i \rVert_{X}. \end{equation*} Then, the dual of $\prod_{i=1}^n X$ is $\prod_{i=1}^n X^*$ with the norm \begin{equation*} \lVert (x_1^*, \ldots, x_n^*) \rVert := \Big( \sum_{i = 1}^n \lVert x_i^* \rVert_{X^*}^q \Big)^{1/q}, \end{equation*} where $q$ is the conjugate exponent to $p$, i.e., $1/p + 1/q = 1$. In case $q = \infty$, i.e., $p = 1$, this is to be understood as \begin{equation*} \lVert (x_1^*, \ldots, x_n^*) \rVert := \max_{i = 1,\ldots,n} \lVert x_i^* \rVert_{X^*}. \end{equation*}