every subspace of a normed space of finite dimension is closed

Let (V,) be a normed vector spacePlanetmathPlanetmath, and SV a finite dimensional subspaceMathworldPlanetmath. Then S is closed.


Let aS¯ and choose a sequencePlanetmathPlanetmath {an} with anS such that an convergesPlanetmathPlanetmath to a. Then {an} is a Cauchy sequencePlanetmathPlanetmath in V and is also a Cauchy sequence in S. Since a finite dimensional normed space is a Banach spaceMathworldPlanetmath, S is completePlanetmathPlanetmathPlanetmathPlanetmathPlanetmath, so {an} converges to an element of S. Since limits in a normed space are unique, that limit must be a, so aS.


The result depends on the field being the real or complex numbers. Suppose the V=Q×R, viewed as a vector space over Q and S=Q×Q is the finite dimensional subspace. Then clearly (1,2) is in V and is a limit point of S which is not in S. So S is not closed.


On the other hand, there is an example where Q is the underlying field and we can still show a finite dimensional subspace is closed. Suppose that V=Qn, the set of n-tuples of rational numbers, viewed as vector space over Q. Then if S is a finite dimensional subspace it must be that S={x|Ax=0} for some matrix A. That is, S is the inverse imagePlanetmathPlanetmath of the closed setPlanetmathPlanetmath {0}. Since the map xAx is continuousPlanetmathPlanetmath, it follows that S is a closed set.

Title every subspace of a normed space of finite dimensionMathworldPlanetmathPlanetmath is closed
Canonical name EverySubspaceOfANormedSpaceOfFiniteDimensionIsClosed
Date of creation 2013-03-22 14:56:28
Last modified on 2013-03-22 14:56:28
Owner Mathprof (13753)
Last modified by Mathprof (13753)
Numerical id 12
Author Mathprof (13753)
Entry type Theorem
Classification msc 54E52
Classification msc 15A03
Classification msc 46B99