Let $I=[0,1]$ and let $\displaystyle X:=\left\{f: I\times \mathbb R\to \mathbb R\colon \sup_{(t,x)}\frac{|f(t,x)|}{1+|x|}<\infty\right\}$. Prove that $X$, equipped with the norm $\displaystyle \|f\|:=\sup_{(t,x)}\frac{|f(t,x)|}{1+|x|}$ is a Banach space.
My first attempt was to use the characterization that $X$ is a Banach space if and only if every absolutely convergent series converges, but no success.
Then I've noticed that I can prove convergence on every Ball of arbitrarily large radius, however still i cannot conclude on the whole $I\times \mathbb R$. Maybe Ascoli Arzela, but, honestly, i don't know.
Hope you can help me. Thank you.