$L_2$ norm is defined as
$$ ||x||_2 = \sqrt{\sum_{i=1}^{n} x_{i}^2} $$
or
$$ ||x||_2 = \sqrt{x^{T}x} $$ (linear operators form)
or
$$ \langle u,v\rangle_2 = \sum_{i=1}^{N} u_{i}v_i $$ (dot product of two vectors that defines norm $L_2$ if $u=v$)
and $$ ||\phi||_{2} = \sqrt{\int\ |\phi(x)^2|dx} $$ but from definition of integal we know $$\int\ |\phi(x)^2|dx = \lim_{n\to\infty}\sum_{i=0}^{n}|\phi(x_{i})\phi(x_{i})|\delta x_{i} $$ then $$ ||\phi||_{2} = \sqrt{\lim_{n\to\infty}\sum_{i=0}^{n}|\phi(x_{i})\phi(x_{i})|\delta x_{i}}$$ that does not correspond to definition #1 if $\phi(x_{i}) = 1\cdot x_{i}$
So the confusion comes form: I do not understand where does the term $\delta x_{i} $ vanishes in the first (or third definitions compared to the last one).
In vector case we take dot product of vector with itself and it gives us its norm, however if we say that the integral above can serve as a definition of $L_2$ norm I do not understand why do not why multiply the dot product by the third vector $\delta X_{i} $. I understand that the scalar product already gives us the scalar (we do not have to multiply it by vector once again) but I do not understand how we obtain a consistency between the definition of integral and $L_2$ norm. Maybe it comes from normalization? The integral form of inner product . But then I cannot see why the boundaries for the norm are not between $0$ and $1$ all the time. Thank you!
