every finite dimensional normed vector space is a Banach space

Theorem 1.

Every finite dimensional normed vector spacePlanetmathPlanetmath is a Banach spaceMathworldPlanetmath.

Proof. Suppose (V,) is the normed vector space, and (ei)i=1N is a basis for V. For x=j=1Nλjej, we can then define


whence :V is a norm for V. Since all norms on a finite dimensional vector space are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath (http://planetmath.org/ProofThatAllNormsOnFiniteVectorSpaceAreEquivalent), there is a constant C>0 such that


To prove that V is a Banach space, let x1,x2, be a Cauchy sequencePlanetmathPlanetmath in (V,). That is, for all ε>0 there is an M1 such that

xj-xk<ε,for allj,kM.

Let us write each xk in this sequencePlanetmathPlanetmath in the basis (ej) as xk=j=1Nλk,jej for some constants λk,j. For k,l1 we then have

xk-xl 1Cxk-xl

for all j=1,,N. It follows that (λk,1)k=1,,(λk,N)k=1 are Cauchy sequences in . As is completePlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath, these convergePlanetmathPlanetmath to some complex numbers λ1,,λN. Let x=j=1Nλjej.

For each k=1,2,, we then have

x-xk Cx-xk

By taking k it follows that (xj) converges to xV.

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