compactness of closed unit ball in normed spaces


finite dimensional \PMlinkescapephraseinfinite-dimensional \PMlinkescapephrasesubspaceMathworldPlanetmathPlanetmath \PMlinkescapephraseconvergent

Theorem - Let X be a normed spaceMathworldPlanetmath and B1(0)¯X the closed ballPlanetmathPlanetmath ( Then B1(0)¯ is compactPlanetmathPlanetmath if and only if X is finite dimensional (

The above result is false, in general, if one is considering other topologiesMathworldPlanetmathPlanetmath in X besides the norm topology (see, for example, the Banach-Alaoglu theorem). It follows that infinite dimensional ( normed spaces are not locally compact.


  • () This is the easy part. Since X is finite dimensional it is isomorphic to some n (with the standard topology). The result then follows from the Heine-Borel theorem.

  • () Suppose that X is not finite dimensional. Pick an element x1X such that x1=1 and denote by S1 the subspace ( generated by x1.

    Recall that, according to the Riesz Lemma (, there exists an element x2X such that x2=1 and d(x2,S1)12, where d(y,M):=inf{y-z:zM} denotes the distance between an element yX and a subspace MX.

    Now consider the subspace S2 generated by x1,x2. Since X is infinite dimensional S2 is a proper subspace and we can still apply the Riesz Lemma to find an element x3 such that x3=1 and d(x3,S2)12.

    If we proceed inductively, we will find a sequence {xn} of norm 1 elements and a sequence of subspaces Sn:=x1,,xn such that d(xn+1,Sn)12. Under this setting it is easily seen that the sequence {xn} is in B1(0)¯ and satisfies xn-xm12 for all mn. Therefore, {xn} is a sequence in B1(0)¯ that has no convergent ( subsequence, i.e., B1(0)¯ is not compact.

Remarks on the proof - Note that the sequence {xn} constructed in the proof does not have a Cauchy subsequence ( Thus we have in fact proven the slightly stronger result that X is finite dimensional if and only if every boundedPlanetmathPlanetmathPlanetmathPlanetmath sequence in X has a Cauchy subsequence.

For Hilbert spacesMathworldPlanetmath the proof would be slightly simpler because one could just pick any orthonormal basisMathworldPlanetmath {en}, and it would en-em=2 for all m,n with mn, therefore having no convergent subsequence. For general normed spaces we cannot just pick orthonormal elements, since this notion does not exist. Thus, we have to use Riesz Lemma to assure the existence of elements with some .

Title compactness of closed unit ball in normed spaces
Canonical name CompactnessOfClosedUnitBallInNormedSpaces
Date of creation 2013-03-22 17:48:41
Last modified on 2013-03-22 17:48:41
Owner asteroid (17536)
Last modified by asteroid (17536)
Numerical id 13
Author asteroid (17536)
Entry type Theorem
Classification msc 46B50
Synonym closed unit ball in a normed space is compact iff the space is finite dimensional