MHB Metric spaces and normed spaces

[ozkan12]

What is the relation between metric spaces and normed spaces... What is the meaning of " metric spaces are seen as a nonlinear version of vector spaces endowed with a norm" ? Thank you for your attention...Best wishes...:)

 

[Ackbach]

All normed spaces are metric spaces: $d(x,y)=\lVert x-y\rVert$ works as a fine metric. But not all metric spaces are normed spaces.

 

[ozkan12]

Dear Ackbach,

I know this...But what is the nonlinearity ? I have troubles related to this term...?

 

[Ackbach]

Dear Ackbach,

I know this...But what is the nonlinearity ? I have troubles related to this term...?

Ah, well, let's see. https://people.math.osu.edu/gerlach.1/math/BVtypset/node7.html has a fine example of a metric space that is not a linear normed space. Does that help?

 

[facenian1]

What is the relation between metric spaces and normed spaces... What is the meaning of " metric spaces are seen as a nonlinear version of vector spaces endowed with a norm" ? Thank you for your attention...Best wishes...:)

Normed spaces are vector spaces while metric spaces are more general. May be that's why the term nonlinear is used.