Norm of a function ||f|| & the root mean square of a function.

[squenshl]

norm of a function ||f|| & the "root mean square" of a function.

How do I explain the connection between the norm of a function ||f|| & the "root mean square" of a function. You may like to consider as an example C[0,$$\pi$$], the inner product space of continuous functions on the interval [0,$$\pi$$] with the inner product
(f,g) = $$\int_0^\pi$$ f(x)g(x) dx.
Let f(x) = sin(x). How do I find ||f||. Also find "root mean square" of f. What do you notice?

 

[morphism]

Well, have you tried using the definitions and applying them to this particular f?

 

[HallsofIvy]

Morphism's point is that |f| is normally defined, in an inner product space, as the square root of the inner product of f with itself. So if
$$(f, g)= \int_0^\pi f(x)g(x)dx$$
Then
$$|f|= \sqrt{\int_0^\pi f^2(x) dx$$

Now, what is
$$\sqrt{\int_0^\pi sin^2(x) dx$$?