Difference between revisions of "Basis"
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When the topological space is a [[vector|vector space]] V, the basis generates V under the [[continuous]] operations of [[addition]] and [[scalar]] multiplication. The [[dimension]] of V is given by the number of elements in the basis, up to intersection and multiplicity. | When the topological space is a [[vector|vector space]] V, the basis generates V under the [[continuous]] operations of [[addition]] and [[scalar]] multiplication. The [[dimension]] of V is given by the number of elements in the basis, up to intersection and multiplicity. | ||
| − | [[category:topology]][[category:algebra]] | + | [[category:topology]][[category:linear algebra]] |
Revision as of 20:27, June 10, 2008
Basis is a mathematics term.
A basis B for a topology T on a set X is a collection of subsets of X (called basis elements) such that
- each x in X, is in at least one basis element.
- if x is in the intersection of 2 basis elements B1 and B2, then it is in some basis element B3, where B3 is a subset of B1 ∩ B2.
If B satisfy the above 2 conditions, then the topology T generated by B is the collection of subsets U of X such that for each x in U, there is a basis element V in B such that x is in V and V is a subset of U.
When the topological space is a vector space V, the basis generates V under the continuous operations of addition and scalar multiplication. The dimension of V is given by the number of elements in the basis, up to intersection and multiplicity.
