Continuity w.r.t a norm -- question about terms Given a set of all continuous functions from the interval $[a,b]$ to the real numbers $\mathbb{R}$, i.e. $C([a,b],\mathbb{R}).$
Let $G :C([a,b],\mathbb{R}) \rightarrow \mathbb{R} , f \mapsto \max_{s \in [a,b]}(f(s))$
and
Let $ \lVert \cdot \rVert_{\infty} :C([a,b],\mathbb{R}) \rightarrow \mathbb{R}, f \mapsto \lVert f \rVert_{\infty}:= \max_{s \in [a,b]}(|f(s)|)$
Show that $G$ is continuous with respect to the norm $ \lVert \cdot \rVert_{\infty} $.
My question hereto is: how should I understand the last sentence, specifically the part "with respect to"? Is it possible to apply the norm $ \lVert \cdot \rVert_{\infty} $ to $G$, and if so, what would the mapping look like? 
Both $G$ and  $ \lVert \cdot \rVert_{\infty} $ are maps from $C([a,b],\mathbb{R}) \rightarrow \mathbb{R}$, hence I am unsure of how to use $G$ and  $ \lVert \cdot \rVert_{\infty} $ together. To do it as "$ \lVert G\rVert_{\infty} $" is not intuitive to me. 
Any help is appreciated. Oh, and thanks for your considerations, smart mathematicians!
 A: In a domain of $G$ we have the $\|\cdot\|_{\infty}$ norm, so continuity wrt. this norm means that if $\|f_n-f\|_{\infty}\to 0$ (as $n\to\infty$) then $|G(f_n)-G(f)|\to 0$. Unfortunately our functional $G$ is not linear because the maximum property loses after multiplying by negative scalars. So, it is not enough to check continuity at $0$-function.
A: We are to take the distance between two functions $f,g$ to be $\|f-g\|_\infty.$ That is implied by the mention of continuity "with respect to" that norm.
The distance between two real numbers $a,b$ is $|a-b|.$
"Continuity" of $G$ means:
\begin{align}
& \text{For every member } f \text{ of the domain } C([a,b],\mathbb{R}), \\[5pt]
& \text{for every number } \varepsilon>0 \text{ (no matter how small)}, \\[5pt]
& \text{there is some number } \delta>0 \text{ so small that whenever} \\[5pt]
& g \text{ is any member of the domain } C([a,b], \mathbb R) \text{ for which} \\[5pt]
& \text{the distance between } f \text{ and } g \text{ is less than } \delta, \\[5pt]
& \text{the distance between } G(f) \text{ and } G(g) \text{ is less than } \varepsilon.
\end{align}
More tersely,
\begin{align}
\forall f\in C([a,b],\mathbb R)\ \forall \varepsilon > 0\  \exists\delta > 0 & \  \forall g\in C([a,b],\mathbb R)\\[10pt]
& \Big( \text{if } \underbrace{\|f-g\|_\infty} < \delta \text{ then } \underbrace{|G(f) - G(g)|}<\varepsilon \Big).
\end{align}
The two $\underbrace{\text{underbraces}}$ correspond to the metric on the domain and the metric on the codomain.
To prove this, you will of course need the definition of $G$. And some basic inequalities.
