How does a norm differ from an absolute value?

1. May 18, 2006

dimensionless

How does a norm differ from an absolute value? For example, is

$$\|\mathbf{x}\| = \sqrt{x_1^2 + \cdots + x_n^2}$$

any different than

$$|\mathbf{x}| = \sqrt{x_1^2 + \cdots + x_n^2}$$

??

2. May 18, 2006

shmoe

The usual absolute value satisfies all the properties of a norm- it's just the most common example of one. You've surely met other examples of norms though.

3. May 18, 2006

Geekster

What the norm is kind of depends on how you are defining the inner product. The example you have is for a normal dot product, but the norm for an inner product is the sqrt of the inner product....

A better definition
http://mathworld.wolfram.com/VectorNorm.html

Last edited: May 18, 2006
4. May 18, 2006

shmoe

Every inner product leads to a norm, but not all norms come from an inner product.

5. May 19, 2006

HallsofIvy

Staff Emeritus
Strictly speaking, the term "absolute value" is used only for numbers: real or complex. The term "norm" is used for vectors. The norm is exactly the same as what you are calling absolute value but I wouldn't use that term.