every subspace of a normed space of finite dimension is closed
Let and choose a sequence with such that converges to . Then is a Cauchy sequence in and is also a Cauchy sequence in . Since a finite dimensional normed space is a Banach space, is complete, so converges to an element of . Since limits in a normed space are unique, that limit must be , so .
The result depends on the field being the real or complex numbers. Suppose the , viewed as a vector space over and is the finite dimensional subspace. Then clearly is in and is a limit point of which is not in . So is not closed.
On the other hand, there is an example where is the underlying field and we can still show a finite dimensional subspace is closed. Suppose that , the set of -tuples of rational numbers, viewed as vector space over . Then if is a finite dimensional subspace it must be that for some matrix . That is, is the inverse image of the closed set . Since the map is continuous, it follows that is a closed set.
|Title||every subspace of a normed space of finite dimension is closed|
|Date of creation||2013-03-22 14:56:28|
|Last modified on||2013-03-22 14:56:28|
|Last modified by||Mathprof (13753)|