Euclidean space (original) (raw)

1 Definition

Euclidean n-space is a metric space (E,d)with the property that the group of isometries is transitive and isisomorphic to an n-dimensional Euclidean vector space. To be more precise, we are saying that there exists an n-dimensional Euclidean vector space V with inner product ⟨⋅,⋅⟩ and amapping

such that the following hold:

1. 1. For all x,y∈E there exists a unique u∈V satisfying

      y=x+u,d⁢(x,y)2=⟨u,u⟩,

2. 1. For all x,y∈E and all u∈V we have

      d⁢(x+u,y+u)=d⁢(x,y).

3. 1. For all x∈E and all u,v∈V we have

      (x+u)+v=x+(u+v).

Putting it more succinctly: V acts transitively and effectively onE by isometries.

Remarks.

• •
  The difference between Euclidean space and a Euclidean vector space is one of loss of structure. Euclidean space is a Euclidean vector space that has “forgotten” its origin.
• •
  A 2-dimensional Euclidean space is often called a_Euclidean plane_.