Definition:Euclidean Space/Real

Definition

Let $\R^n$ be an $n$-dimensional real vector space.

Let the Euclidean metric $d$ be applied to $\R^n$.

Then $\struct {\R^n, d}$ is a Euclidean $n$-space.

Also see

• Euclidean Metric on Real Vector Space is Metric

• Results about Euclidean spaces can be found here.

Source of Name

This entry was named for Euclid.

Historical Note

Euclid himself did not in fact conceive of the Euclidean metric and its associated Euclidean space, Euclidean topology and Euclidean norm.

They bear that name because the geometric space which it gives rise to is Euclidean in the sense that it is consistent with Euclid's fifth postulate.

Sources

• 1963: Louis Auslander and Robert E. MacKenzie: Introduction to Differentiable Manifolds ... (previous) ... (next): Euclidean, Affine, and Differentiable Structure on $R^n$: $\text {1-1}$: Euclidean $n$-Space, Linear $n$-Space, and Affine $n$-Space
• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $2$: Continuity generalized: metric spaces: $2.2$: Examples: Example $2.2.1$
• 1999: Theodore W. Gamelin and Robert Everist Greene: Introduction to Topology (2nd ed.) ... (previous) ... (next): One: Metric Spaces: $1$: Open and Closed Sets: $(1.5)$