TITLE: Proving $\left\lVert f\right\rVert_{\infty}:=\sup_{x \in X}\left\lVert f(x)\right\rVert_2$ is finite and the norm of a continuous subset
QUESTION [0 upvotes]: I'm trying to answer this Real Analysis question below,

Let $K$ be a compact subset of $R^2$, and let $C(K,\mathbb R^m)$ denote the vector space of all continuous functions from $K$ into $\mathbb R^m$. Show that for $f$ in $C(K,\mathbb R^m)$ , the quantity $\left\lVert f\right\rVert_{\infty}:=\sup_{x \in X}\left\lVert f(x)\right\rVert_2$ is finite and $\left\lVert 
 \cdot\right\rVert_\infty$ is the norm on $C(K,\mathbb R^m)$.

I have absolutely no idea how to answer this question. I have a very basic understand for the Uniform Norm and a very basic understanding of compactness (I understand the definitions and nothing else). All I can figure out about this question is how to state the expansion of a two-norm,
$\left\lVert f(x)\right\rVert_2=(\sum_{n=1}^{m}|f_n(x)|^2)^{1/2}$
Could I please receive some hints on how I'm supposed to solve this question? I'm extremely incapable with abstract mathematics so keeping the hints very simple would help a lot too.

REPLY [0 votes]: What kind of a function is $\left\|f(x)\right\|_2$? What do you know about such functions on compact domains?
{"set_name": "stack_exchange", "score": 0, "question_id": 3974570}