Norm of sine\cosine

Question

In $L_2$ space we define norm as dot product on itself, for sinus it will be: $$ ||\sin(x)|| = \left[\int_{0}^{2\pi}\sin(x)\sin(x)dx\right]^{1/2}$$

And after simple integrating we find that:

$$||\sin(x)|| = \sqrt{\pi}$$

so my question is:

in terms of Euklidian space we deem that norm it's a length\amplitude of vector, but what does it mean when we concern about functions, not vectors?

P.S. I hope i could explain my thoughts properly

Answer 1

To a rather strong extent, the answer is “it’s still the magnitude of the vector. You’re not just good ad visualizing Hilbert Spaces.” Your final sentence is wrong; $L^2$ is a vector space, and $\sin(x)$ is a vector inside of that vector space.