How does a norm differ from an absolute value?

[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

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

??

 

[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.

 

[Geekster]

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

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

??

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

 

[shmoe]

What the norm is kind of depends on how you are defining the inner product.

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

 

[HallsofIvy]

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.