$L^p$ norm of $g:\mathbb{R}\rightarrow\mathbb{R}^n$

Question

If I understand correctly, the $L^p$ norm of $f:\mathbb{R}\rightarrow\mathbb{R}$ is $$\|f\|_p=\left(\int_{-\infty}^\infty |f(x)|^pdx\right)^{1/p}$$

But is the $L^p$ norm of $g:\mathbb{R}\rightarrow\mathbb{R}^n$ defined?

Is it this? $$\left(\int_{-\infty}^\infty \|g(x)\|_pdx\right)^{1/p}$$ Is there a name for this norm? $$\left(\int_{-\infty}^\infty \|g(x)\|_2^{p}dx\right)^{1/p}$$

Answer 1

The space of all measurable functions $g: \mathbb R \to \mathbb R^{n}$ with $\int \|g\|_2^{p}<\infty$ is denoted by $L^{p} (\mathbb R, \mathbb R^{n})$. More generally we can talk about $L^{p} (\mathbb R , X)$ for any normed linear space $X$.

However $(\int\|g\|_p)^{1/p}$ is not a norm.