Metric spaces and normed spaces

  • Context: MHB 
  • Thread starter Thread starter ozkan12
  • Start date Start date
  • Tags Tags
    Metric
Click For Summary

Discussion Overview

The discussion revolves around the relationship between metric spaces and normed spaces, exploring the meaning of the characterization of metric spaces as a nonlinear version of vector spaces endowed with a norm. Participants are seeking clarification on the concept of nonlinearity in this context.

Discussion Character

  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant states that all normed spaces are metric spaces, providing the metric defined by the norm, but notes that not all metric spaces are normed spaces.
  • Another participant expresses confusion about the term "nonlinearity" and seeks further explanation.
  • A later reply references an example of a metric space that is not a linear normed space, suggesting it may help clarify the concept of nonlinearity.
  • One participant reiterates the relationship, indicating that normed spaces are vector spaces while metric spaces are more general, which may explain the use of the term nonlinear.

Areas of Agreement / Disagreement

Participants generally agree that normed spaces are a subset of metric spaces, but there is uncertainty and lack of consensus regarding the meaning of nonlinearity and its implications.

Contextual Notes

Participants express confusion about specific terminology and concepts, particularly regarding nonlinearity, without resolving these uncertainties.

ozkan12
Messages
145
Reaction score
0
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...:)
 
Physics news on Phys.org
Dear Ackbach,

I know this...But what is the nonlinearity ? I have troubles related to this term...?
 
ozkan12 said:
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?
 
ozkan12 said:
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.
 

Similar threads

Undergrad Open balls dense in closed balls in Euclidean space
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Normed Vector Spaces and Topological Vector Spaces
  • · Replies 9 ·
Replies
9
Views
3K
MHB Norm bounded Sets .... remarks by Garling in Section 11.2 Normed Spaces ....
  • · Replies 2 ·
Replies
2
Views
2K
MHB Uniform norm .... Garling Section 11.2 Normed Saces .... also Example 11.5.7
  • · Replies 4 ·
Replies
4
Views
2K
MHB Bounded in Norm .... Garling, Section 11.2: Normed Spaces ....
  • · Replies 5 ·
Replies
5
Views
2K
Undergrad Bounded in Norm .... Garling, Section 11.2: Normed Spaces ....
  • · Replies 2 ·
Replies
2
Views
3K
Undergrad How Do Metric Space Properties Relate to Continuity?
  • · Replies 8 ·
Replies
8
Views
2K
MHB How Do We Know the Annihilator Operation Is Associative?
  • · Replies 2 ·
Replies
2
Views
1K
MHB Normed and Inner Product Spaces .... Garling, Corollary 11.3.2 ....
  • · Replies 2 ·
Replies
2
Views
1K
MHB Isometries .... Garling, Example 11.5.2 .... ....
  • · Replies 2 ·
Replies
2
Views
1K