Definition Suppose is a vector space and suppose that is a non-empty subset of . If there exists a such that is a vector subspace of , then is a linear manifold of . Then we say that the dimension of is the dimension of and write . In the important case , is called a hyperplane.
A linear manifold is, in other words, a linear subspace that has possibly been shifted away from the origin. For instance, in examples of linear manifolds are points, lines (which are hyperplanes), and itself. In hyperplanes naturally describe tangent planes to a smooth hyper surface.
- 1 R. Cristescu, Topological vector spaces, Noordhoff International Publishing, 1977.
|Date of creation||2013-03-22 14:04:32|
|Last modified on||2013-03-22 14:04:32|
|Last modified by||matte (1858)|