Understanding Vector Spaces: ||x||_inf and max |x_i| in R^n

[squaremeplz]

Homework Statement

Does ||x||_inf = max | x_i | for 1 <= i <= n define a norm on R^(n)

Homework Equations

The Attempt at a Solution

ok, I thought I understood vector spaces but this problem is confusing the heck out of me.

A norm is a function that assigns a positive and finite length to all vectors in a vector space.

so ||x||_inf = sqrt(x1^2 + x2^2 + ... x_inf)

max |x_i| depends on n in R^n

Can someone give me like a simple example? i.e. n = 2

Then ||x||_inf = max (|x1|, |x2|)?

The maximum distance between all vectors would be equal to the distance from 0 to the greatest vector. rather, infinity would be considered bounded by the max vector?

Any help is greatly appreciated.

 

[upsidedowntop]

To determine if a function is a norm, you must check if is satisfies three axioms for all v,w in the vector space V:

||v|| >= 0 and equal to 0 iff v = 0.
|| v + w || <= || v || + || w ||
|| cv || = |c| || v ||,
where c is any scalar in the field the vector space is over.

Good Luck!

 

[squaremeplz]

So the the question is asking whether max |x_i| for 1 <= i <= n satisfies these conditions?

 

[HallsofIvy]

Your definition of the norm is wrong. I think what you are writing (but [itex]x_{inf} should be squared and you should write $\sum_{i=1}^\infty x_i^2$- there is no "$x_{inf}$" member) is the "Euclidean Norm" (as you titled this thread) for $l_n$, the space of square summable sequences. The "infimum" norm on Rⁿ is $MAX(x_1, x_2, \cdot\cdot\cdot, x_n)$.

 

[squaremeplz]

Thanks.