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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?

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    $\begingroup$ Not many would like to download the doc. Please type out the definition of the space and the norm. $\endgroup$ – Kavi Rama Murthy Apr 22 at 5:29
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    $\begingroup$ The norm is defined on slide 5, in terms of the example inner product defined on slide 4. The specific formula is written on slide 6, in terms of integrals, just above the claim about $\sqrt{3}x$. It's an abstract definition, trying to make a sensible notion of distance in an infinite-dimensional space. It won't have anything to do with arc length, or any other measure of distance evident in the graphs of the functions. $\endgroup$ – user771918 Apr 22 at 5:30
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    $\begingroup$ In this abstraction, length is not measuring anything specific, but is simply a synonym of 'norm'. $\endgroup$ – Berci Apr 22 at 10:46
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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.

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    $\begingroup$ +1. $C^0([0,1])$ is often denoted by $C[0,1].$ $\endgroup$ – DanielWainfleet Apr 22 at 11:50

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