# Elementary proof of $C^\infty_c$ is dense in $L^p (L^q)$ mixed space

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?

• Your idea looks good: just approximate these $a_i$'s by smooth functions, and you get a smooth approximation of $f\in L^p(L^q)$. – daw Jul 19 '16 at 7:09