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.

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All labels observed (7)

Label	Occurrences
Euclidean space canonical	11
Euclidean geometry	2
Euclidean spaces	2
Euclidean plane	1
Euclidean plane has K = 0	1
Euclidean space-time	1
Euclidean spaces Rn	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 (19)

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

Riemannian manifolds → generalizes → Euclidean space ⓘ

subject surface form: Riemannian manifold

Newtonian mechanics → assumes → Euclidean space ⓘ

Whitney embedding theorem → concerns → Euclidean space ⓘ

Galilean relativity → assumes → Euclidean space ⓘ

Gaussian curvature → exampleConstantCurvatureSurface → Euclidean space ⓘ

this entity surface form: Euclidean plane has K = 0

Cartesian coordinate system → geometryType → Euclidean space ⓘ

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

this entity surface form: Euclidean geometry

Erlangen Program → classifies → Euclidean space ⓘ

this entity surface form: Euclidean geometry

Schwinger functions → spaceTimeDomain → Euclidean space ⓘ

Osterwalder–Schrader axioms → assumes → Euclidean space ⓘ

this entity surface form: Euclidean space-time

Pythagorean theorem → holdsIn → Euclidean space ⓘ

Euclidean group → actsOn → Euclidean space ⓘ

Lie sphere geometry → appliesTo → Euclidean space ⓘ

Newtonian celestial mechanics → assumes → Euclidean space ⓘ

Cauchy–Schwarz inequality → appliesTo → Euclidean space ⓘ

this entity surface form: Euclidean spaces

Fourier inversion theorem → holdsIn → Euclidean space ⓘ

this entity surface form: Euclidean spaces Rn

Conway's thrackle conjecture → ambientSpace → Euclidean space ⓘ

this entity surface form: Euclidean plane

Monge problem in optimal transport → domain → Euclidean space ⓘ

this entity surface form: Euclidean spaces

Penrose triangle → cannotExistIn → Euclidean space ⓘ