Difference between revisions of "Talk:Euclidean space"
From Encyclopedia of Mathematics
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The phrase "<u>In a more general sense</u>, a Euclidean space is a finite-dimensional real vector space" does not sound right. Since a finite dimensional real vector space satisfies the abstract Euclidean axioms, it is <u>less</u> general, not more. The general sense should be that, on adding suitable axioms such as continuity and between-ness to those of Euclid, one obtains a system which more or less characterises the real vector space. [[User:Richard Pinch|Richard Pinch]] ([[User talk:Richard Pinch|talk]]) 20:02, 3 March 2018 (CET) | The phrase "<u>In a more general sense</u>, a Euclidean space is a finite-dimensional real vector space" does not sound right. Since a finite dimensional real vector space satisfies the abstract Euclidean axioms, it is <u>less</u> general, not more. The general sense should be that, on adding suitable axioms such as continuity and between-ness to those of Euclid, one obtains a system which more or less characterises the real vector space. [[User:Richard Pinch|Richard Pinch]] ([[User talk:Richard Pinch|talk]]) 20:02, 3 March 2018 (CET) | ||
| + | :I guess, "more general" means here "not necessarily of dimension 2 or 3". Axioms of Euclid are too ancient... rather, one uses [[Hilbert system of axioms]] or another system that is ''equivalent'' to real vector space with inner product in dimensions 2 and 3. [[User:Boris Tsirelson|Boris Tsirelson]] ([[User talk:Boris Tsirelson|talk]]) 21:34, 3 March 2018 (CET) | ||
Revision as of 20:34, 3 March 2018
More or less generality
The phrase "In a more general sense, a Euclidean space is a finite-dimensional real vector space" does not sound right. Since a finite dimensional real vector space satisfies the abstract Euclidean axioms, it is less general, not more. The general sense should be that, on adding suitable axioms such as continuity and between-ness to those of Euclid, one obtains a system which more or less characterises the real vector space. Richard Pinch (talk) 20:02, 3 March 2018 (CET)
- I guess, "more general" means here "not necessarily of dimension 2 or 3". Axioms of Euclid are too ancient... rather, one uses Hilbert system of axioms or another system that is equivalent to real vector space with inner product in dimensions 2 and 3. Boris Tsirelson (talk) 21:34, 3 March 2018 (CET)
How to Cite This Entry:
Euclidean space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Euclidean_space&oldid=42902
Euclidean space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Euclidean_space&oldid=42902