every finite dimensional normed vector space is a Banach space
Theorem 1.
Every finite dimensional normed vector space is a Banach space
.
Proof. Suppose is the normed vector space, and is a basis for . For , we can then define
whence is a norm for .
Since
all norms on a finite dimensional vector space are equivalent (http://planetmath.org/ProofThatAllNormsOnFiniteVectorSpaceAreEquivalent),
there is a constant such that
To prove that is a Banach space, let be a Cauchy sequence
in . That is,
for all there is an such that
Let us write each in this sequence in the basis
as for some constants
.
For we then have
for all .
It follows that
are Cauchy sequences in . As is complete, these converge
to
some complex numbers .
Let .
For each , we then have
By taking it follows that converges to .
Title | every finite dimensional normed vector space is a Banach space |
---|---|
Canonical name | EveryFiniteDimensionalNormedVectorSpaceIsABanachSpace |
Date of creation | 2013-03-22 14:56:31 |
Last modified on | 2013-03-22 14:56:31 |
Owner | matte (1858) |
Last modified by | matte (1858) |
Numerical id | 10 |
Author | matte (1858) |
Entry type | Theorem |
Classification | msc 46B99 |