# Difference between revisions of "Talk:Euclidean space"

From Encyclopedia of Mathematics

(Created page with "==More or less generality== 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 dime...") |
(→More or less generality: not necessarily of dimension 2 or 3) |
||

Line 1: | Line 1: | ||

==More or less generality== | ==More or less generality== | ||

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 21:34, 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://www.encyclopediaofmath.org/index.php?title=Euclidean_space&oldid=42897