tensor product and dual spaces

Let k be a field and V be a vector spaceMathworldPlanetmath over k. Recall that

V*={f:Vk|f is linear}

denotes the dual spacePlanetmathPlanetmath of V (which is also a vector space over k).

PropositionPlanetmathPlanetmath. Let V and W be vector spaces. Consider the map ϕ:V*W*(VW)* such that


Then ϕ is a monomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath. Moreover if one of the spaces V, W is finite dimensional, then ϕ is an isomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath.

Proof. One can easily check that ϕ is a well defined linear map, thus it is sufficient to show that Ker(ϕ)=0. So assume that FV*W* is such that ϕ(F)=0. It is clear that F can be (uniquely) expressed in the form


where (fi) is a basis of V*, (gj) is a basis of W* and αi,jk. Then for any vV and wW we have:


Since wW is arbitrary then we can write this equality in the form:


and since (gj) are linearly independentMathworldPlanetmath we obtain that iαi,jfi(v)=0 for all j. Again since vV was arbitrary we obtain that iαi,jfi=0 for all j. Now since (fi) are linearly independent we obtain that αi,j=0 for all i,j. Thus F=0.

Now assume that dimkV=q<+. Let (vi)i=1q be a basis of V and let (vi*)i=1q be an induced basis of V*. Moreover let (wp)pP be a basis of W. We wish to show that ϕ is onto, so let f:VWk be an element of (VW)*. Define FV*W* by the formulaMathworldPlanetmathPlanetmath:


where gi:Wk is such that gi(wp)=f(viwp). Then for any vj from (vi)i=1q and for any wp from (wp)pP we have:


and thus ϕ(F)=f.

Remark. The map ϕ from the previous proposition is very important in studying algebrasMathworldPlanetmathPlanetmath and coalgebras (more precisly it is an essence in defining dual (co)algebras). Unfortunetly ϕ does not have to be an isomorphism in general. Nevertheless, the spaces (VW)* and V*W* are always isomorphic (see this entry (http://planetmath.org/TensorProductOfDualSpacesIsADualSpaceOfTensorProduct) for more details).

Title tensor productPlanetmathPlanetmathPlanetmath and dual spaces
Canonical name TensorProductAndDualSpaces
Date of creation 2013-03-22 18:31:51
Last modified on 2013-03-22 18:31:51
Owner joking (16130)
Last modified by joking (16130)
Numerical id 6
Author joking (16130)
Entry type Theorem
Classification msc 15A69