Should you drop the inner absolute value sign for $L2$ norm?

Question

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?

Answer 1

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 $\lvert x\rvert^{p} = \left(\lvert x\rvert^{2}\right)^{q} = \left( x^{2}\right)^{q} = x^{2q} = x^p$.