Unit ball weakly* compact

Question

So here is my problem,

I am trying to understand the proof of,

$X$ Banach space $\Rightarrow$ the unit ball in $X^*$ is weakly* compact.

The proof uses Tychonoffs Theorem to conclude the compacness. I think I undestand the whole proof except the following step,

Define $K_x:=[-||x||,||x||]\subset\mathbb R$ then we have,

$$\prod_{x\in X}K_x=\{f:X\rightarrow\mathbb R\;|\;|f(x)|\leq||x||\;\forall x \in X\}$$ and I really dont see why those sets are equal, can someone help me? Thanks!!

Answer 1

An application $$ f:X\to \Bbb R $$can be considered as $$ (f(x))_{x\in X} \in \Bbb R^X $$