What is $L^1$ norm for $n$-fold functional space of integrable function?

Question

Consider a space of $n$-fold real-valued integrable functions, $X \doteq [L^1([0,T])]^n$. Some use an alternative notation like $X \doteq L^1([0,T]; \mathbb{R}^n)$.

1. Does $f\in X$ mean $f_1\in[L^1([0,T])]$, $f_2\in[L^1([0,T])]$, ..., $f_n[L^1([0,T])]$?

2. In the context of the Lebesgue dominated convergence theorem, what does $|f(t)|\leq\psi(t)$ mean? Does it mean $|f_i(t)| \leq \psi_i(t)$ for all $i=1,2,..,n$?

3. What is the $L^1$ norm for $X$? It seems $\int_0^T |f(t)| dt$, which is unclear to me. $f$ is a vector, then how can it be integrated? Is it something like $\int_0^T |f(t)|^\top dt$ with $dt$ being also a vector?

I'm studying practical applications and having hard time to apply these concepts.

Answer 1

If we follow all those delicacies in $L^p$-spaces theory, then the answers are:

1. Yes.

2. Yes. What you say is equivalent to just say $|f_i(t)| \leq \psi(t)$ pointwisely for just one $\psi$ though. Then it is like applying Dominated Convergence Theorem component-wise for $f_i$.

3. For a vector field $\mathbf{f}= (f_1,\ldots,f_n)\in L^1([0,T];\mathbb{R}^n)$, the $L^1$-norm is: $$ \|\mathbf{f} \|_{L^1([0,T];\mathbb{R}^n)} = \int^T_0 (|f_1(t)|+\cdots+|f_n(t)|)dt \tag{1} $$ the value of the norm should be a scalar.

UPDATE: The $L^1$-integrable vector field is a special case for Bochner space, the value can be in a Banach space $(B,\|\cdot\|_B)$. In your case, this space is $(\mathbb{R}^n,\|\cdot\|_{l^p})$, for all little $p$ norm, $\mathbb{R}^n$ is Banach. So here it really depends on what norm you are using for $\mathbb{R}^n$, in the answer I used $l^1$-norm by default: for a number $a\in \mathbb{R}^n$ $$ \|a\|_{l^1} = |a_1| + \cdots +|a_n| $$ If the $\mathbb{R}^n$ is equipped with $l^2$-norm that measures the traditional Euclidean distance: $$ \|a\|_{l^2} = (|a_1|^2 + \cdots +|a_n|^2)^{1/2} $$ Then the $L^1$-norm for $L^1([0,T];\mathbb{R}^n)$ is: $$ \|\mathbf{f} \|_{L^1([0,T];\mathbb{R}^n)} = \int^T_0 (|f_1(t)|^2+\cdots+|f_n(t)|^2)^{1/2}dt \tag{2} $$ Both (1) and (2) is $\displaystyle \int^T_0 \|\mathbf{f}(t)\|_{l^p} dt$ in the sense that $L^1$-integrability is for time variable.

Choosing which one to use depends on the contexts. If the optimal control problem has discrete state variables, I am highly inclined to use (1) which measures the Manhattan distance between states at each time $t$.

The reference can be found in Yosida's functional analysis, look for "Bochner integral" in the index.

BTW: I am curious of what practical applications you are using these concepts in.