every finite dimensional subspace of a normed space is closed
Proof. Let be such a normed vector space, and a finite dimensional vector subspace.
Let , and let be a sequence in which converges to . We want to prove that . Because has finite dimension, we have a basis of . Also, . But, as proved in the case when is finite dimensional (see this parent (http://planetmath.org/EverySubspaceOfANormedSpaceOfFiniteDimensionIsClosed)), we have that is closed in (taken with the norm induced by ) with , and then . QED.
The definition of a normed vector space requires the ground field to be the real or complex numbers. Indeed, consider the following counterexample if that condition doesn’t hold:
is a - vector space, and is a vector subspace of . It is easy to see that (while is infinite), but is not closed on .
|Title||every finite dimensional subspace of a normed space is closed|
|Date of creation||2013-03-22 14:58:56|
|Last modified on||2013-03-22 14:58:56|
|Last modified by||Mathprof (13753)|