MATH 511A, HOMEWORK 4
Let σbe an automorphism of F, and let V, W be F-vector spaces. The goal of
this homework assignment is to prove the following theorem.
Theorem. Let h,i:V×W→Fbe a perfect σ-sesquilinear pairing of finite-
dimensional vector spaces. If `∈V∗, there exists w∈Wsuch that `(v) = hv, wi
for all v∈V.
Recall that a function f:V→Wis called σ-linear if f(v1+v2) = f(v1) +
f(v2) and f(cv) = σ(c)f(v). Let the set of σ-linear maps V→Wbe denoted
Homσ(V, W ).
1.
(a) Check that Homσ(V, W ) is actually an F-vector space, where (f1+f2)(v) =
f1(v) + f2(v) and (a·f)(v) = af(v).
(b) If {vi}are a basis for Vand {wi}are arbitrary elements of W, show that
there exists a unique element f∈Homσ(V, W ) with f(vi) = wifor all i.
(c) Define a map T: Homσ(V, W )→Hom(V , W ) as follows: if f∈Homσ(V, W ),
then T(f) is the unique linear map sending vi7→ f(vi) for all i. Prove
that Tis an isomorphism of vector spaces. In particular deduce that
dim Homσ(V, W ) = dim Hom(V , W ).
2. Suppose that f∈Homσ(V, W ).
(a) If fis injective, prove that dim(V)≤dim(W).
(b) If fis surjective, prove that dim(V)≥dim(W).
(c) If fis a bijection, prove that dim(V) = dim(W).
3. Suppose h,i:V×W→Fis a pairing.
(a) We obtain a map φ:W→V∗sending wto the map v7→ hv, wi. Show
that φis σ-linear.
(b) Similarly we obtain a map ψ:V→Homσ(W, F) sending vto the map
w7→ hv, wi. Show that ψis linear.
4.
(a) If h,i:V×W→Fis a perfect pairing and either Vor Wis finite-
dimensional, prove that dim(V) = dim(W).
(b) Give an example to show that if V, W are infinite-dimensional and h,i:
V×W→Fis a perfect pairing, then it need not be the case that dim(V) =
dim(W).
5. Prove the theorem.
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