Euclidean space

E22816

affine space geometric space inner product space mathematical concept metric space normed vector space topological space

Euclidean space is the standard flat, n-dimensional geometric setting of classical geometry and vector calculus, characterized by straight lines, right angles, and the usual distance and dot product.

Jump to: Surface forms Statements Referenced by

Observed surface forms (3)

Surface form	Occurrences
Euclidean geometry	2
Euclidean plane has K = 0	1
Euclidean space-time	1

Statements (50)

Predicate	Object
instanceOf	affine space ⓘ geometric space ⓘ inner product space ⓘ mathematical concept ⓘ metric space ⓘ normed vector space ⓘ topological space ⓘ
generalizedBy	Riemannian manifold ⓘ
geodesicsAre	straight lines ⓘ
hasAngleDefinition	via dot product ⓘ
hasBasis	orthonormal basis ⓘ
hasCoordinateSystem	Cartesian coordinates ⓘ
hasCurvature	zero ⓘ
hasDimension	n ⓘ
hasDistanceFunction	Euclidean distance ⓘ
hasFieldOfScalars	real numbers ⓘ
hasIsometryGroup	E(n) ⓘ
hasMetric	Euclidean metric ⓘ
hasNorm	Euclidean norm ⓘ
hasOperation	dot product ⓘ scalar multiplication ⓘ vector addition ⓘ
hasStandardBasis	canonical basis of R^n ⓘ
hasStraightLines	geodesics ⓘ
hasStructure	vector space over the real numbers ⓘ
hasSubspace	affine subspaces ⓘ lines ⓘ planes ⓘ
hasSymmetryGroup	Euclidean group ⓘ
hasTopology	standard Euclidean topology ⓘ
isComplete	true ⓘ
isConnected	true ⓘ
isFlat	true ⓘ
isHausdorff	true ⓘ
isHomogeneous	true ⓘ
isLocallyCompact	true ⓘ
isPathConnected	true ⓘ
isSecondCountable	true ⓘ
isSeparable	true ⓘ
isSimplyConnected	true ⓘ
namedAfter	Euclid ⓘ
satisfies	Pythagorean theorem ⓘ parallelogram law ⓘ triangle inequality ⓘ
specialCaseOf	Hilbert space ⓘ
standardModel	R^n ⓘ
usedIn	classical geometry ⓘ classical mechanics ⓘ multivariable calculus ⓘ vector calculus ⓘ

Referenced by (10)

Full triples — surface form annotated when it differs from this entity's canonical label.

Galilean relativity → assumes → Euclidean space ⓘ

Newtonian mechanics → assumes → Euclidean space ⓘ

Osterwalder–Schrader axioms → assumes → Euclidean space ⓘ

this entity surface form: Euclidean space-time

Erlangen Program → classifies → Euclidean space ⓘ

this entity surface form: Euclidean geometry

Whitney embedding theorem → concerns → Euclidean space ⓘ

Gaussian curvature → exampleConstantCurvatureSurface → Euclidean space ⓘ

this entity surface form: Euclidean plane has K = 0

Riemannian manifolds → generalizes → Euclidean space ⓘ

subject surface form: Riemannian manifold

Cartesian coordinate system → geometryType → Euclidean space ⓘ

Über die Hypothesen, welche der Geometrie zu Grunde liegen → influencedBy → Euclidean space ⓘ

this entity surface form: Euclidean geometry

Schwinger functions → spaceTimeDomain → Euclidean space ⓘ