I was given an exercise, where I'm asked to calculate the second normal of a difference between a function $f(x)$ and it's approximation $p(x)$:
$||f-p||_2$
I find this notation to be very confusing, because how can we take the Euclidean norm of a function? (assuming I can mark $f(x)-p(x)=g(x)$ )
In the previous exercises we used the infinite norm for functions, which was defined as $||f(x)||_\infty=max\space |f(x)|$
I was guessing that the valid way to convert it to a continuous operation is to replace the summation in the definition of the norm ($||x||_2=\sqrt{\sum_{i=1}^n{|x_i|}}$) to an integral, but I am not sure as to how that is supposed to be done.
Also as a side note, we have this form used as a way to write down the Pythagorean formula, and it looks like so:
$||\varphi''||_2^2=||s_n''||_2^2+||\varphi''-s_n''||_2^2\geq||s_n''||_2^2$
I find it hard to relate such algebraic expressions to the seemingly geometric nature of both the Euclidean normal form and the Pythagorean formula and would appreciate any help