closure of a vector subspace is a vector subspace
Let be the topological vector space over where is either or , let be a vector subspace in , and let be the closure of . To prove that is a vector subspace of , it suffices to prove that is non-empty, and
whenever and .
First, as , contains the zero vector, and is non-empty. Suppose are as above. Then there are nets , in converging to , respectively. In a topological vector space, addition and multiplication are continuous operations. It follows that there is a net that converges to .
We have proven that , so is a vector subspace. ∎
|Title||closure of a vector subspace is a vector subspace|
|Date of creation||2013-03-22 15:00:19|
|Last modified on||2013-03-22 15:00:19|
|Last modified by||loner (106)|