What is the Infinity Norm & Why Use It?

[alecrimi]

Hi
I was wondering about the meaning of the infinity norm
$|| x ||_\inf= max\{|x_1|, |x_2|...|x_n| \}$

if a norm is a function that assigns a strictly positive length or size to all vectors in a vector space, why do we assign the maximum (or sup) as the value of this norm ?
It must be a very basic or obvious answer because I cannot find in any text.

 

[uart]

I'm not sure of exactly what you're asking alerimi.

It is positive (and zero only when the entire vector is identically zero). It's very easy to show that it satisfies the triangle inequality. Exactly what problem do you have with it's use as a norm?

 

[alecrimi]

why is it defined by "max" ?

 

[uart]

why is it defined by "max" ?

Do you mean in the sense of limit of the sequence one-norm, two-norm etc?

Ok I think that must be what you're asking, what is the connection between infinity in the name and maximum in the definition.

There are a family of norms,

$$|| x ||_1 = |x_1| + |x_2| + ... |x_n|$$
$$|| x ||_2 = \left( |x_1|^2 + |x_2|^2 + ... |x_n|^2 \right)^{\frac{1}{2}}$$
$$|| x ||_p = \left( |x_1|^p + |x_2|^p + ... |x_n|^p \right)^{\frac{1}{p}}$$

Imagine that you take the limit as p goes to infinity. Factor out the largest |x_i| and then the i-th element becomes unity, while the all other elements are (magnitude) less than one. Think about what happens as you take the p-th power of each element now (as p -> infinity). That's an oversimplification, but hopefully it let's you see how it works.

 

[alecrimi]

yes, it was that. I thought it was something like that. Thanks