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What is the precise definitions of "Euclidean space" and "Space"?

This question was asked in my viva-exam i'm not sure about my answer-I answered like-"Euclidean space is a space where the notion of Euclidean distance is present",then the examiner counter questioned me "What is a space?" to this i've no answer.

I need to know how should i answered these questions.I've also gone through wikipedia for "Euclidean space",but i did'nt get any precise answer.

Any suggestions are heartly welcome!!

Picaso
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  • "space" is not a very lucky word. There are many "spaces" in mathematics : Banach-spaces, vector-spaces, Hilbert-spaces,... You should have said "An Euclidean space is a vector space ..." . It is precisely defined what a vector-space is. Just google it. In a related question someone mentioned that a "space" is a "set with some structure", but this vague as well because what is meant with "some structure" ? – Peter Aug 17 '17 at 10:38
  • @Peter: Concerning the definition of Euclidean spaces: An Euclidean space is a normed vector space where the norm is the Euclidean norm. The Euclidean norm is the so called 2-norm .This i get Xaver's comment in https://math.stackexchange.com/questions/1803141/the-precise-definition-of-cartesian-coordinate-and-euclidean-space – Picaso Aug 17 '17 at 10:45
  • @Peter:After giving some time to this notion of $space$,i reached to this statement(it may be wrong,Please check )$SPACE-$Space is some structue formed from the elements of any certain set following some laws. – Picaso Aug 17 '17 at 10:51
  • Coincides with what I heard, but to be honest, I do not know the definition of a "space". I only know definitions of special spaces, for example the definition of a vector space. – Peter Aug 17 '17 at 10:53
  • @Peter:According to your link it is "page not found". – Picaso Aug 17 '17 at 11:10
  • The question was deleted. Therefore I removed the link. – Peter Aug 17 '17 at 11:11
  • @Peter:May be!! – Picaso Aug 17 '17 at 11:12
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    While it can be an interesting philosophical question what a "space" is, my guess is that your examiner simply wanted you to make precise what kind of space you meant. As other commenters have said, the correct definition of Euclidean space is that it is a special kind of vector space. – mrp Aug 17 '17 at 12:02

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The answer really depends on the context. If the question was in the context of a Euclidean geometry class and was about the Euclidean plane, one could give Hilbert's list of axioms, including his primitive notions. (I am actually not even sure about a reference for a Hilbert-type list of axioms for the Euclidean 3-space.) If it was in the context of a linear algebra class, the right answer would be to give axioms of an $n$-dimensional real vector space equipped with an inner product, but regarded as an affine space rather than a vector space (i.e. undergone the "affinization"). If this were in the context of a metric geometry class, one would give yet another definition and state that the $n$-dimensional Euclidean space $E^n$ is a metric space satisfying a certain list of properties which I do not want to give here. Lastly, if this were in the context of a differential geometry class, the answer would be that $E^n$ is the (isometry class of) the $n$-dimensional complete simply connected Riemannian manifold of zero curvature.

Moishe Kohan
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