Canonical Isomorphism Between a Finite Dimensional Inner Product Space and its Dual
Point of Post: In this post we prove that every finite dimensional inner product space is isomorphic to its dual space.
We have seen in the past the proof that every finite dimensional vector space is isomorphic to its double dual. We know of course since dimension is preserved under taking duals for finite dimensional vector spaces (this is, in fact, a characterization of finite dimensionality) but there was no canonical (free of basis choice) way of defining the mapping. In this post we prove the scene is different if the vector space is supplied with an inner product (or more generally a non-degenerate bilinear form).
We recall that a bilinear form for some finite dimensional -space is one such that and are never the zero map unless or are themselves . With this in mind we prove that:
Theorem: Let be a finite dimensional -space and a distinguished non-degenerate bilinear form on . Then, the map
,where , is an isomorphism.
Proof: Clearly this map is linear since for every and one has that
and so . Moreover, by definition of non-degeneracy we have that and thus by a common theorem regarding linear transformations and so is injective. Thus, we have that and so (recalling our previous comment about finite dimensional vector spaces being isomorphic to their duals) from where it follows that . The conclusion follows.
Remark: It’s clear that is also an isomorphism.
Where this most comes up is the following:
Corollary: Let be a finite dimensional inner product space over (where ) then every linear functional is of the form for some fixed , and moreover this is unique.
1. Roman, Steven. Advanced Linear Algebra. New York: Springer-Verlag, 1992. Print.