Is it true that the absolute value $|\cdot|$ is equivalent to all $l_p$ norms?

Question

It seems to me that every $l_p$ norm reduces to the absolute value norm $|\cdot|$ in $\mathbb{R}$, i.e. $||x||_1 = ||x||_2 = \ldots = ||x||_\infty, x \in \mathbb{R}$

Wikipedia treats the absolute value norm as a special case of $l_1$ norm.

What is the correct treatment here?

Answer 1

The answer is yes. If you are restricting yourself to $\mathbb{R}$ then since the definition is $||x||_p=(|x_1|^{p})^{1/p}=|x_1|,$ the reduction you are asking about does hold. Differences only begin to appear in 2 or more dimensions.