MHB Metric spaces and normed spaces

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Normed spaces are a specific type of metric space where the distance is defined by a norm, making all normed spaces inherently metric spaces. However, not all metric spaces can be classified as normed spaces due to their broader nature. The term "nonlinear" refers to the fact that metric spaces do not necessarily adhere to the linear structure found in vector spaces. An example of a metric space that is not a linear normed space illustrates this distinction. Understanding these differences clarifies the relationship between 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.
 

Thread 'About the definition of topological manifold using closed sets'

We all know the definition of n-dimensional topological manifold uses open sets and homeomorphisms onto the image as open set in ##\mathbb R^n##. It should be possible to reformulate the definition of n-dimensional topological manifold using closed sets on the manifold's topology and on ##\mathbb R^n## ? I'm positive for this. Perhaps the definition of smooth manifold would be problematic, though.
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