Proof of continuity of norm in function space

Question

How would you go about proving that the norm induced by the scalar product $$ \int_a^b f\cdot g \mathop{dx}$$
in the space $\mathcal{C}([a,b])$ is continuous? Trying to calculate $$ \left|\int_a^b |f|^2 \mathop{dx} - \int_a^b |g|^2 \mathop{dx} \right|$$ it is clear that I can find an upper bound for this quantity provided that $f$ and $g$ are "close enough", but what notion of proximity should I use? The same norm? Wouldn't the reasoning be circular?

Answer 1

The norm which induces the topology is always continuous (for the usual topology on $\mathbb{R}$) since you can write that

$$x=x-y+y\text{ so }||x||\leq ||x-y||+||y||\text{ and then }||x||-||y||\leq ||x-y||,$$

and also

$$||y||-||x||\leq ||x-y||,$$

which leads to

$$\big|||x||-||y||\big|\leq||x-y||,$$

so the norm is $1$-lipschitz and so continuous.