# What is the difference between the notations $\|x\|_1, \|x\|_2$ and $|x|$?

What is the difference between the notations $$\|x\|_1, \|x\|_2$$ and $$|x|$$? I think $$|x|$$ is the magnitude of $$x$$.

## 1 Answer

$$\|x\| = |x|$$ denotes the absolute value norm, which is a special case of the $$L_1$$ norm defined on the 1-D vector spaces formed by real or complex numbers.

$$\|\textbf{x}\|_1 = \sum_{i=1}^n|x_i|$$ denotes the Taxicab / Manhattan norm, relating to how a Taxi would drive along a rectangular grid of roads to reach a point $$(x, y)$$ from $$(0,0)$$.

$$\|\textbf{x}\|_2 = \sqrt{x_1^2 + \dots + x_n^2}$$ denotes the Euclidean norm on an N-D Euclidean space, which is a result of the Pythagorean theorem (the shortest distance between two points).

• Could you please elucidate on absolute value norm computation. Jan 27, 2020 at 5:18
• The Absolute value norm is a special case of the $L_1$ norm (defined in 1D). So, computing it would be equivalent to evaluating the absolute value of the single component of the vector.
– s_bh
Jan 27, 2020 at 10:54