As it well known, $C^\infty_0 (\mathbb{R}^n)$(the space of infinitely differentiable functions with compact support) is dense in $L^p (\mathbb{R)^n}$
Here I want to consider the same result with mixed norm. The mixed norm space is defined by $$ \Vert f \Vert_{L^p_t L^q_x}:=\left(\int_{\mathbb R} \left[\int_{\mathbb{R}^n} |f(t,x)|^p dx\right]^{\frac{q}{p}}dt\right)^{\frac{1}{p}}.$$
This norm can be viewed as $$ \Vert f \Vert_{L^p_t L^q_x}:=\left(\int_{\mathbb R} \Vert f(t,\cdot) \Vert_{L^q (\mathbb{R^n})}^p dt \right)^{\frac{1}{p}}.$$
So by this point of view, I can verify that the space $L^p_t L^q_x$ is actually $L^p (\mathbb{R}; L^q(\mathbb{R}^n)$) space, which is a special case of Bochner space.
Then in the theory of Banach-valued function space, $$ f(t) = \sum_{i=1}^n a_i \phi_i (t),$$ where $a_i$ is an element of Banach space $B$, $\phi_i$ is a characteristic function in $\mathbb{R}$ with finite measure, is dense in $L^p (\mathbb{R};B)$ when $1\leq p <\infty$. So by taking molification, we see that $C^\infty_0 (\mathbb{R};B)$ is dense in $L^p (\mathbb{R};B)$.
As $C^\infty_0$ is dense in $L^p(\mathbb{R}^n)$, $C^\infty_0 (\mathbb{R}^{n+1})$ is dense in $L^p_t L^q_x$.
My question is the following:
Can I proved it properly?
How can prove the fact without using Bochner integral?
Thank you in advance.