tensor product of dual spaces is a dual space of tensor product

PropositionPlanetmathPlanetmathPlanetmath. Let k be a field and V, W be vector spacesMathworldPlanetmath over k. Then (VW)* is isomorphic to V*W*.

Proof. If V or W is finite dimensional, then there is an explicit isomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath between (VW)* and V*W* (see this entry (http://planetmath.org/TensorProductAndDualSpaces) for more details). So assume that both V and W are infinite dimensional.

First of all, note that if V is a vector space, then dimk(V) denotes its dimensionPlanetmathPlanetmath, that is dimk(V) is a cardinality of any basis of V. Thus we can compare dimensions of spaces, so dimk(V)dimk(W) if and only if there is an injection from a basis of V to a basis of W (note that this relationMathworldPlanetmathPlanetmath is well defined, i.e. it does not depend on the choice of bases). One can easily show that dimk(V)dimk(W) if and only if there is an injective linear map from V to W and this is if and only if there is a surjectivePlanetmathPlanetmath linear map from W to V. Therefore V is isomorphic to W if and only if dimk(V)=dimk(W) (which here means that dimk(V)dimk(W) and dimk(W)dimk(V)).

Without loss of generality, we may assume that dimk(V)dimk(W) (we can always compare any two sets). Note that the basis of VW is the productMathworldPlanetmathPlanetmathPlanetmath of bases of V and W (namely if (ei)iI is a basis of V and (ej)jJ a basis of W, then (eiej)iI,jJ is a basis of VW). If X,Y are sets such that Y is infiniteMathworldPlanetmath and there is an injection XY, then it is well known that there is a bijection from Y to X×Y. Thus (since W is infinite dimensional) we have:


Therefore VW is isomorphic to W, so (VW)* is isomorphic to W*.

Now the inequality dimk(V)dimk(W) implies that dimk(V*)dimk(W*). Indeed, assume that there is an injective linear map f:VW. Then there is a (surjective) linear map g:WV such that gf=idV (see this entry (http://planetmath.org/SomeFactsAboutInjectiveAndSurjectiveLinearMaps) for more details). Therefore (since ()* is a contravariant functorMathworldPlanetmath) we have that


and this implies that f*:W*V* is a surjective linear map (on the other hand g*:V*W* is an injective linear map), so dimk(V*)dimk(W*).

Now (due to previous arguments) we have


so V*W* and W* are isomorphic.

All in all, we have that (VW)* is isomorphic to W*, which is isomorphic to V*W*. This completesPlanetmathPlanetmathPlanetmathPlanetmath the proof.

Remarks. We know that there is an isomorphism between (VW)* and V*W*, but generally we know nothing about it, about its behaviour. Thus it is hard to find imprortant applications for this proposition. Also note, that this proposition is true for free modules over any unital ring.

Title tensor productPlanetmathPlanetmathPlanetmath of dual spacesPlanetmathPlanetmath is a dual space of tensor product
Canonical name TensorProductOfDualSpacesIsADualSpaceOfTensorProduct
Date of creation 2013-03-22 18:32:19
Last modified on 2013-03-22 18:32:19
Owner joking (16130)
Last modified by joking (16130)
Numerical id 18
Author joking (16130)
Entry type Theorem
Classification msc 15A69