Norm of a vector-valued function?

Solution 1:

When you have a function $v \colon X \to E$ on a measure space $X$ with values in a Banach space $E$, saying that $v\in L^p(X,\,E)$ usually means that $v$ is a measurable function such that the real-valued function $n \colon x \mapsto \lVert v(x)\rVert$ is in $L^p(X,\,\mathbb{R})$, and $\lVert v\rVert_p = \left(\int \lVert v(x)\rVert_E^p\right)^{1/p}$.

In the case that $E = \mathbb{R}^d$, firstly, that is equivalent to each component function itself being in $L^p$, and secondly, since all norms on $\mathbb{R}^d$ are equivalent, it doesn't matter which norm we choose on $\mathbb{R}^d$, they all produce equivalent norms on $L^p(X,\,\mathbb{R}^d)$. You then get the most convenient computations if you choose the corresponding $\ell^p$-norm on $\mathbb{R}^d$, which becomes

$$\lVert v\rVert_p = \left(\int_X \sum_{i=1}^d \lvert v_i(x)\rvert^p \,d\mu\right)^{1/p},$$

the $p$-th power of the norm of $v$ is the sum of the $p$-th powers of the norms of the components.