Assertion: euclidean space's dimension | /r/HasProperty | finite
Facets:
value: coincide in this case
type: OTHER
value: for an algebra over a field
type: OTHER
value: with rational coefficients(in other words
type: OTHER
Source sentences:
text: Euclidean spaces have finite dimension.
text: Finally one could generalize and allow the dimension to be countably infinite, and regard separable Hilbert spaces as generalized Eclidean spaces.
text: Finite examples in dimension 2 (finite affine planes) have been valuable in the study of configurations in infinite affine spaces, in group theory, and in combinatorics.
text: Euclidean spaces have finite dimension.
text: with rational coefficients(in other words, if α is algebraic ), the dimension is finite.
text: Now, the conditions in the list are what you likely known (perhaps without the complex 'decorations') the conditions for a Euclidean space where there is the additional condition that the dimension is finite.
text: For an algebra over a field, the dimension as vector space is finite if and only if its Krull dimension is 0.
text: This dimension is finite if and only if the space's Hamel dimension is finite, and in this case the two dimensions coincide.
text: An Euclidean Space has finite dimensions
text: If no such integer n exists, then the dimension of X is said to be infinite, and one writes dim X = ∞.