What does a "norm in the space of continuous functions" look like?

Question

On page 3 of this document, the norm in the space of continuous functions is defined, and then there is an example given that the length of $\sqrt3x$ is $1$. What does "length" mean when talking about the "space of continuous functions?" It cannot be the arc length, because the arc length on $[0, 1]$ of that function is $2$. The definition of norm as a length makes sense, but what length is it measuring?

Answer 1

For simplicity, let us define space of continuous functions $X:=C^0([a,b])$ as : $$C^0([a,b]):=\left\{f:[a,b] \rightarrow \mathbb{R} : \text{f is continous} \right\}$$ You can define norm on $X$ in many ways, one of the most common one is

$$ C^0([a,b]) \ni u \mapsto \Vert u \Vert_{\infty} := \sup \limits_{x \in [a,b]} \vert u(x) \vert \in \mathbb{R}_{\ge 0} $$ Note $\left( C([a,b]),|| \cdot ||_{\infty} \right)$ is a Banach space.

The one described in the document you talk about, is the standard $L^2$ norm i.e $$C^0([a,b]) \ni u \mapsto \Vert u\Vert_2 :=\left(\int_{\mathbb{R}}|u(x)|^2 \mathrm{d}x \right)^{1/2}=: \langle u,u \rangle^{1/2} \in \mathbb{R}_{\ge 0}$$ Check: $\left(C^0([a,b]),||\cdot ||_2 \right)$ is a Banach space or not ?

So you just need to apply this definition to $\sqrt{3}x$ in your question. You can think the $L^2$ distance as infinite dimensional analogue of euclidean distance but it doesn't have to be, it's a more abstract definition.