Determining whether these functions are norms

[Ezequiel]

Homework Statement

Determine whether these functions are norms:

a. ||.||_2/3: ℝⁿ → ℝ
||(x₁,...,x_n)||_2/3 = (Ʃ|x_i|^2/3)^3/2.

b. If ||.|| is a norm and λ \in ℝ\{0}
||x||_λ = λ||x||.

c. If ||.||₁ and ||.||₂ are norms:
||x||_max = max{||x||₁,||x||₂}.

d. If ||.||₁ and ||.||₂ are norms:
||x||_min = min{||x||₁,||x||₂}.

Homework Equations

A norm on a vector space X is a real-valued function on X whose value at an x \in X is denoted by ||x|| and which has the properties:

1. ||x|| ≥ 0
2. ||x|| = 0 ⇔ x = 0
3. ||αx|| = |α| ||x||
4. ||x + y|| ≤ ||x|| + ||y||

here x and y are arbitrary vectors in X and α is any scalar.

The Attempt at a Solution

What I would do is check if the functions have the four properties listed above.

I think that ||.||_λ is not a norm because λ||x|| can be negative.

I know that ||.||_2/3 has the first three properties, but I don't know how to check if it has the fourth one.

And as for ||.||_max and ||.||_min, I don't even know what max{} and min{} mean.

Any help would be appreciated! Thanks.

 

[gopher_p]

What I would do is check if the functions have the four properties listed above.

Good.

I think that ||.||_λ is not a norm because λ||x|| can be negative.

That's right. Is that the only thing that goes wrong? What if we stipulate that λ must be positive (instead of just nonzero)?

I know that ||.||_2/3 has the first three properties, but I don't know how to check if it has the fourth one.

What are the easiest vectors to check? The second easiest?

And as for ||.||_max and ||.||_min, I don't even know what max{} and min{} mean.

If you had to guess, what would say that max{} and min{} meant? Seriously, take a guess. I can almost guarantee you that you'll be right (assuming you're a native speaker of English).

 

[Ezequiel]

That's right. Is that the only thing that goes wrong? What if we stipulate that λ must be positive (instead of just nonzero)?

#1 ||x||_λ ≥ 0 ⇔ λ > 0 (λ \in ℝ\{0})

#2 ||x||_λ = 0 ⇔ x = 0 (since λ ≠ 0 and ||x||_λ = λ||x||)

#3 ||αx||_λ = λ ||αx|| = λ |α| ||x|| = |α| ||x||_λ

#4 ||x+y||_λ ≤ ||x||_λ + ||y||_λ

λ ||x+y|| ≤ λ (||x|| + ||y||)

and ||x+y|| ≤ (||x|| + ||y||) holds because ||.|| is a norm.

Thus, ||.||_λ must be a norm for λ > 0. Am I right?

What are the easiest vectors to check? The second easiest?

I don't know. Can you tell me? :)

For ||.||_2/3 #1 and #2 are trivial.

#3 ||αx||_2/3 = (Ʃ|αx_i|^2/3)^3/2 = (Ʃ|α|^2/3|x_i|^2/3)^3/2 = (|α|^2/3 Ʃ |x_i|^2/3)^3/2 = |α| (Ʃ |x_i|^2/3)^3/2 = |α| ||x||_2/3

I don't know how to check #4:

(Ʃ |x_i+y_i|^2/3)^3/2 ≤ (Ʃ |x_i|^2/3)^3/2 + (Ʃ |y_i|^2/3)^3/2

If you had to guess, what would say that max{} and min{} meant? Seriously, take a guess. I can almost guarantee you that you'll be right (assuming you're a native speaker of English).

Well, even though English is not my native language I'm quite sure max and min stand for maximum and minimum respectively. I'm just not very familiar with them as mathematical symbols. However, if they mean what I think they mean, I suppose that #1 and #2 are trivial for ||.||_max and ||.||_min.

I'm not sure about #3, but I think that

||αx||_max = max{||αx||₁,||αx||₂} = max{|α| ||x||₁,|α| ||x||₂} = |α| max{||x||₁,||x||₂} = |α| ||x||_max

and the same for ||.||_min.

I don't know how to check #4 for these functions either :(