Lp norm is defined as:
$ \left\| \mathbf{x} \right\| _p := \bigg( \sum_{i=1}^n \left| x_i \right| ^p \bigg) ^{1/p}$
But often time I see people writing:
$\left\| \mathbf{x} \right\| _2 := \bigg( \sum_{i=1}^n x_i^2 \bigg) ^{1/2}$
for $L_2$ norm
When is it acceptable to do this? Just for $L_2$ norm?
For any $L_p$ norm where $p$ is even, and the $x_i$'s are real numbers. (In all generality, the $L_p$ norm can be defined for vectors in $\mathbb{C}$, for which you do need to keep the "absolute values" -- except that they are the modulus).
This directly follows from the fact that $\lvert x\rvert^2 = x^2$, when $x\in\mathbb{R}$. So that if $p=2q$ for some integer $q$, then $\lvert x\rvert^{p} = \left(\lvert x\rvert^{2}\right)^{q} = \left( x^{2}\right)^{q} = x^{2q} = x^p$.
(1) To begin with, the symbol ":=" is often used quite subtly to demarcate a definition in addition to an equality, rather than an equality alone [1].
(2)
For $L_2$, It is acceptable to write the latter in place of the former when all the $x_i$ (i = 1, 2, ... n) are real numbers.
(3)
For finite, even-valued $p$, it is acceptable to write the latter for all $L_p$ norms when all the $x_i$ (i = 1, 2, ... n) are real numbers.
(4)
For finite $p$, it is acceptable to write the latter for all $L_p$ norms when all the $x_i$ (i = 1, 2, ... n) are non-negative numbers.
Bibliography:
[1] What is the distinction between $=$ and $:=$ when writing math?
