F(x)=||x|| is Lipschitz function

In summary, the function f(x)=||x|| is a Lipschitz function from a normed vector space into [0,∞) by the triangle inequality and the proof follows the same pattern as in R. In metric spaces, there is also a variant of the triangle inequality that applies to the function d(x,0) and it is always a Lipschitz function.
  • #1
kingwinner
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"Let (V,||.||) be a normed vector space. Then by the triangle inequality, the function f(x)=||x|| is a Lipschitz function from V into [0,∞)."

I don't understand how we this follows from the triangle inequality. How does the proof look like?

Any help is appreciated!
 
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  • #2
[tex]\Big|\|x\| - \|y\| \Big|\le \|x-y\|[/tex]
 
  • #3
OK, I think that's the "variant triangle inequality". I've seen the proof in R (by the regular triangle inequality), but how can we prove it for normed vector spaces?
 
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  • #4
g_edgar said:
[tex]\Big|\|x\| - \|y\| \Big|\le \|x-y\|[/tex]


[tex]\Big|\|x\| - \|y\| \Big|\le \|x-y\|[/tex] <=> -||x-y||[tex]\leq||x||-||y||\leq[/tex] ||x-y|| <=> -||x-y|| +||y||[tex]\leq ||x||\leq[/tex] ||x-y|| +||y|| which is correct since

||x|| =||x-y+y||[tex]\leq[/tex] ||y|| +||x-y|| and

||y|| = ||y-x+x||[tex]\leq[/tex] ||x-y||+ ||x||
 
  • #5
Thanks!

Just wondering...is there a analogue of the "variant triangle inequality" in general METRIC SPACES? If so, how does it look like and how does the proof change?
 
  • #6
kingwinner said:
Thanks!

Just wondering...is there a analogue of the "variant triangle inequality" in general METRIC SPACES? If so, how does it look like and how does the proof change?
[tex]\left|d(x,0)-d(y,0)\right|\leq d(x,y)[/tex]

The same proof
 
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  • #7
evagelos said:
[tex]\left|d(x,0)-d(y,0)\right|\leq d(x,y)[/tex]

The same proof

So in the metric space context, does it mean that the function "d" is always a Lipchitz function as well?
 
  • #8
kingwinner said:
So in the metric space context, does it mean that the function "d" is always a Lipschitz function as well?

The Lipschitz function is f(x) =d(x,0) and not d(x,y)
 

1. What is the definition of a Lipschitz function?

A Lipschitz function is a function that satisfies the Lipschitz condition, which means that there exists a positive real number K such that the absolute value of the difference between the function's output at any two points is less than or equal to K times the absolute value of the difference between those two points.

2. Why is it important for a function to be Lipschitz?

Lipschitz functions have a bounded rate of change, meaning that they do not change too quickly or too dramatically. This property is important in many applications, such as in optimization problems, where having a Lipschitz function can guarantee the existence and uniqueness of a solution.

3. How is the Lipschitz constant calculated for a function?

The Lipschitz constant, also known as the Lipschitz bound, can be calculated by taking the maximum absolute value of the derivative of the function over the entire domain. In the case of the function f(x) = ||x||, the Lipschitz constant is equal to 1.

4. Can a function be Lipschitz if it is not continuous?

No, a function must be continuous in order to be Lipschitz. This is because the Lipschitz condition requires that the function's output does not change too much when the input changes by a small amount, and this concept is only applicable to continuous functions.

5. Are all Lipschitz functions differentiable?

No, not all Lipschitz functions are differentiable. For a function to be differentiable, it must be continuous and have a well-defined derivative. While Lipschitz functions are continuous, they may not have a well-defined derivative at every point in their domain.

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