1. Dual Space Let V be a vector space over a field F. A linear functional on V is a linear map f : V → F. The set of all linear functions on V is denoted by V ∗. By definition, V ∗ = L(V,F ). The set L(V,F ) is also a vector space over F.

We assume that V is an n dimensional vector space over F.

Let β = {v1, ··· , vn} be a basis for V. For each v ∈ V, there exist unique numbers a1, ··· , an ∈ F such that v = a1v1 + ··· + anvn. For each 1 ≤ i ≤ n, we define fi(v) = ai. Then fi : V → F defines a linear functional on V for each 1 ≤ i ≤ n. ∗ Definition 1.1. The ordered set β = {f1, ··· , fn} is called the set of coordinate functions on V with respect to the basis β.

It follows from the definition that fi(vj) = δij for 1 ≤ i, j ≤ n.

Theorem 1.1. The set β∗ forms an ordered basis for V ∗ such that for any f ∈ V ∗, n X (1.1) f = f(vi)fi. i=1 ∗ Proof. Let us prove that β is linearly independent over F. Suppose a1f1 + ··· + anfn = 0. By fi(vj) = δij, for each 1 ≤ j ≤ n, one has

(a1f1 + ··· + anfn)(vj) = a1f1(vj) + ··· + anfn(vj) = aj = 0. Now let us prove that β∗ and (1.1) at the same time. Pn ∗ Let f be given. Define g : V → F by g(v) = i=1 f(vi)fi(v). Then g ∈ V . Furthermore, for each 1 ≤ j ≤ n, n n X X g(vj) = f(vi)fi(vj) = f(vi)δij = f(vj). i=1 i=1 We see that the two linear functionals f and g coincide on β. Sine f, g are linear, f = g on ∗ V. We find that f ∈ span β and f has the representation of the form (1.1).  Definition 1.2. The ordered basis β∗ is called the dual basis to β. ∗ Corollary 1.1. dimF V = dimF V . ∗∗ ∗ ∗ Theorem 1.2. Let V be the dual space to V . For each v ∈ V, we define vb : V → F by sending f to f(v). Then (1) vb : V → F is a linear map; ∗∗ (2) the function ϕ : V → V sending v to vb is a linear isomorphism. Proof. We leave it to the reader to check that vb is a linear map.

Let us show that ker ϕ = {0}. Let v ∈ ker ϕ. Then ϕ(v) = 0. Hence vb(f) = 0 for any ∗ f ∈ V . Write v = a1v1 + ··· + anvn for some a1, ··· , an ∈ F. Since f(v) = 0 for all v ∈ V, fi(v) = 0 for all 1 ≤ i ≤ n which implies that ai = fi(v) = 0 for all 1 ≤ i ≤ n. We see that v = 0. Since V ∗∗ is the dual basis to V ∗ and V ∗ is the dual basis to V, by Corollary 1.1, ∗∗ ∗ dimF V = dimF V = dimF V. ∗∗ Since ϕ : V → V is a linear monomorphism (linear and injective) with dimF V = ∗∗ dimF V , ϕ is a linear isomorphism.  1