Metric spaces and normed spaces

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SUMMARY

Metric spaces are general structures that can be viewed as nonlinear versions of vector spaces endowed with a norm. All normed spaces qualify as metric spaces since the distance function defined by the norm, $d(x,y)=\lVert x-y\rVert$, serves as a valid metric. However, not all metric spaces are normed spaces, highlighting the distinction between these two mathematical concepts. The term "nonlinearity" refers to the broader applicability of metric spaces compared to the linear constraints of normed spaces.

PREREQUISITES
  • Understanding of vector spaces
  • Familiarity with norms and distance functions
  • Basic knowledge of metric spaces
  • Concept of linearity versus nonlinearity in mathematics
NEXT STEPS
  • Study the properties of metric spaces in detail
  • Explore examples of metric spaces that are not normed spaces
  • Investigate the implications of nonlinearity in mathematical structures
  • Learn about the applications of normed spaces in functional analysis
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Mathematicians, students of advanced mathematics, and anyone interested in the foundational concepts of metric and normed spaces.

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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...:)
 
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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.
 

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