Functions belonging to $L^p(\mathbb{R}^n ; \mathbb{R}^m)$ if and only of their norm belongs to $L^p(\mathbb{R}^n)$

Question

The definition I have of $f \in L^p(\mathbb{R}^n ; \mathbb{R}^m)$ is that we require each component function to be in $L^p(\mathbb{R}^n)$.

Is is true that $f \in L^p(\mathbb{R}^n ; \mathbb{R}^m) \Longleftrightarrow |f|\in L^p(\mathbb{R}^n)$? Where $| \cdot|$ denotes the usual Euclidean norm.

Answer 1

If $f\in L^p(\mathbb{R}^n;\mathbb{R}^m)$, then $f=(f_1,\ldots, f_m)$ with $f_j\in L^p(\mathbb{R}^n)$ for $j=1,\ldots, m$, therefore $$|f|=\sqrt{f_1^2+\ldots+ f_m^2}\leq \sqrt{n}\max_j |f_j|$$ is in $L^p(\mathbb{R}^n)$ (because the maximum of $L^p$ functions is in $L^p$).

On the other hand, as $$|f_j|\leq\sqrt{|f_1|^2+\ldots+|f_m|^2}=|f|^2$$ if the latter is in $L^p$, also the former is.