The set of all linear functionals, denoted by V∗, forms a vector space

JIA-MING (FRANK) LIOU

Let K be a field. All the vector spaces in this note are over K.

1. Some Linear Algebras

Let V be an n-dimensional vector space over K. A linear functional on V is a linear map ϕ : V → K. The set of all linear functionals, denoted by V^∗, forms a vector space. The addition and scalar multiplication are given as follows. Let ϕ, ψ ∈ V^∗, and a ∈ k. Define ϕ + ψ and aϕ by

(ϕ + ψ)(v) = ϕ(v) + ψ(v), (aϕ)(v) = aϕ(v).

Theorem 1.1. V^∗ has dimension n.

Proof. Let {v1, · · · , vn} be a basis for V. Let ψ_i : V → K such that ψi(vj) = δij. Claim that the set β = {ψ_i: 1 ≤ i ≤ n} forms a basis for V^∗.

For any ϕ ∈ V^∗, one can check that ϕ =

n

X

i=1

ϕ(e_i)ψ_i. Then ϕ is a linear combinaton of elements of β.

Assume that Pn

i=1aiψi = 0 for some a1, · · · , an ∈ K. Evaluate the sum for v_k, we see that a_k= 0. This shows that β is linearly independent.

 Remark. The basis {ψi} for V^∗ in Theorem (1.1) is called the dual basis of {vi}.

Let ϕ₁, ϕ₂ ∈ V^∗. We define a map ϕ₁∧ ϕ₂: V × V → K by (ϕ₁∧ ϕ₂)(w₁, w₂) = det [ϕ_i(w_j)]²_i,j=1.

We can check that ϕ1∧ ϕ₂ : V × V → K is bilinear, and skew-symmetric, i.e. ϕ2∧ ϕ₁ =

−ϕ₁ ∧ ϕ₂. We call ϕ₁∧ ϕ₂ the wedge product of ϕ₁ and ϕ₂. Let Λ²V^∗ be the set of all skew-symmetric bilinear maps from V × V to K.

Theorem 1.2. Λ²V^∗ is a vector space of dimensionn 2

 .

Proof. Let {v_i} be a basis for V and {ψ_i} be its dual basis. Claim that {ψ_i∧ ψ_j : 1 ≤ i <

j ≤ n} forms a basis for V.

Suppose that

X

i<j

aijψi∧ ψ_j = 0.

Evaluate the equation for (vk, vl), we obtain akl= 0.

Let B be a skew-bilinear map. Denote b_ij = B(v_i, v_j). Then b_ij = −b_ji. For any x = Pn

i=1xivi and y =P

iyivi, we see that B(x, y) =X

i,j

bijxiyj 1

=X

i<j

b_ijx_iy_j+X

i>j

b_ijx_iy_j

=X

i<j

b_ij(x_iy_j − x_jy_i).

Note that (ψ_i∧ ψ_j)(x, y) = (x_iy_j − x_jy_i). Hence we find B(x, y) = (X

i<j

bijψi∧ ψ_j)(x, y).

Hence we prove that B =P

i<jb_ijψ_i∧ ψ_j. 

Definition 1.1. Let p be a natural number and Sp be the symmetric group on p-letters.

A map f : ×^p_i=1V → K is alternating if

f (v_σ1, · · · , v_σp) = (sgn σ)f (v₁, · · · , v_p), for any v1, · · · , vp ∈ V. f is called p-linear if for each 1 ≤ k ≤ p,

f (v₁, · · · , av_k+ bw_k, · · · , v_p) = af (v₁, · · · , v_k, · · · , v_p) + bf (v₁, · · · , w_k, · · · , v_p), for any a, b ∈ K.

Let 1 ≤ p ≤ n. Given ϕ₁, · · · , ϕ_p ∈ V^∗, we define ϕ₁∧ · · · ∧ ϕ_p : ×^p_i=1V → k by (ϕ1∧ · · · ∧ ϕ_p)(w1, · · · , wp) = det [ϕi(wj)]^p_i,j=1,

for all w₁, · · · , w_k ∈ V. Then we can check that ϕ₁∧ · · · ∧ ϕ_p is p-linear and alternating.

Let Λ^pV^∗ be the set of all p-linear alternating maps from ×^p_i=1V to k and denote Λ⁰V^∗= V^∗.

Theorem 1.3. For 0 ≤ p, ≤ n, the set Λ^pV^∗ forms a vector space of dimensionn p

 . Proof. Let {v_i} be a basis for V and {ψ_i} be a basis for V^∗ dual to V. We only need to show that the set {ψi1 ∧ · · · ∧ ψ_ip : 1 ≤ i1< · · · < ip ≤ n} forms a basis for Λ^pV^∗.

 2. Tangent Vectors

Let p be a point in Rⁿ. We denote (v)_p = p + v, where + is the addition in Rⁿ. Let T_pRⁿ be the set of all (v)_p with v ∈ Rⁿ. We define the addition and scalar multiplication on T_pRⁿ as follows. Let (v)p, (w)p ∈ T_pRⁿ, and a ∈ R. We define

(v)_p+ (w)_p = (v + w)_p, a · (v)_p= (av)_p.

Note that the addition and scalar multiplication here are different from those of Rⁿ. Proposition 2.1. The set TpRⁿ forms an n-dimensional real vector space.

Proof. The proof is obvious. Let {ei : 1 ≤ i ≤ n} be the standard basis for Rⁿ. Then {(e_i)_p: 1 ≤ i ≤ n} forms a basis for T_pRⁿ.

 Remark. We will use the notation vp for (v)p.

We say that TpRⁿ is the tangent space to Rⁿ at p and elements of TpRⁿ are tangent vectors at p. The dual space of T_pRⁿ is denoted by T^∗Rⁿ and called the cotangent space of Rⁿ at p.

Let U be an open set on Rⁿ. The algebra of (real-valued) smooth functions on U is denoted by C^∞(U ). Let p ∈ U and f ∈ C^∞(U ), we define the directional derivative of f at p along a vector v ∈ Rⁿ by

v_p[f ] = d

dtf (p + tv)    _t=0

.

Let {ei : 1 ≤ i ≤ n} be the standard basis for Rⁿ. Assume that v = Pn

i=1viei. From calculus, we know

vp[f ] =

n

X

i=1

vi

∂f

∂xi

(p).

Then we know

(1) v_p : C^∞(U ) → R is a linear functional, i.e. vp(af + bg) = av_p(f ) + bv_p(g) for all a, b ∈ R and

(2) v_p(f g) = v_p(f )g(p) + f (p)v_p(g).

Note that if the tangent vector is given, the directional derivative only depends on the partial derivatives of the function at the point p. Suppose f, g are two smooth functions defined on some open sets, (not necessarily the same). If f and g agree on an open set containing p, then vp(f ) = vp(g). This leads to another definition of tangent vectors of Rⁿ. Let V, W be open sets containing p in Rⁿ. Let f be a smooth function on V and g smooth function on W. We say that the pair (f, V ) is equivalent to the pair (g, W ) if there exists an open set Z ⊂ V ∩ W contain p such that f = g on Z. The set of all equivalent classes [(f, V )]

is denoted by C_p^∞. For simplicity, we will simply denote [(f, V )] by [f ]. An equivalent class [f ] is called a germ at p.

Proposition 2.2. The set C_p^∞forms an algebra over R.

Let [f ] ∈ C_p^∞. We define [f ](p) = f (p), where f is a representative of [f ]. Then [f ](p) is well-defined. Now, if f1, f2 belong to the same germ [f ] at p, we can check that vp(f1) = v_p(f₂). Hence we can define v_p([f ]) = v_p(f ) where f is a representative of [f ].

Definition 2.1. A point derivation δ_p at p is a linear functional on C_p^∞ such that δ_p([f ][g]) = δ_p([f ])[g](p) + [f ](p)δ_p([g]).

The set of all point derivations at p is denoted by T_p⁰Rⁿ.

Remark. By definition, all tangent vectors are point derivation at p.

Given [f ] ∈ C_p^∞, we define

∂

∂xi

(p)[f ] = ∂f

∂xi

(p) for a representative f of [f ]. Then we know that ∂

∂x_i(p) is a point derivation.

Theorem 2.1. The set

 ∂

∂x_i(p) : 1 ≤ i ≤ p



forms a basis for T_p⁰Rⁿ; Then T_p⁰Rⁿ is an n-dimensional vector space.

Proof. It is easy to verify that the set is linearly independent.

Let δ_p ∈ T_pRⁿ. Denote δ_p[x_i] = v_i. Let [f ] ∈ C_p^∞. Choose a representative f of [f ].

Consider the Tayler expansion f (x) = f (p) +X

i

∂f

∂xi

(p)(xi− p_i) +X

i,j

(xi− p_i)(xj− p_j) Z 1

0

(1 − t) ∂²f

∂xi∂xj

(p + t(x − p))dt.

Using the properties of tangent vectors, we find δ_p[f ] =X

i

v_i ∂

∂xi

(p)[f ].

In other words, we find δp =X

i

vi

∂

∂xi

(p). 

We know that T_pRⁿis a vector subspace of T_p⁰Rⁿ. Since they both have the same dimen- sion, they must be equal. In fact, by elementary calculus,

(e_i)_p[f ] = ∂f

∂x_i(p).

We find that (ei)p = ∂

∂xi

(p) by definition. Hence we conclude that the notion of tangent vectors are equivalent to the notion of point derivations.

Remark. Since we always choose a representative of a germ at p, from now on, we will simply use the notation f for [f ].

Let f be a germ at p. We define a linear functional df_p: T_pRⁿ→ R by dfp(vp) = vp(f )

Then df_p ∈ T_p^∗Rⁿ. Notice that for any v_p = (v₁, · · · , v)_p ∈ T_pRⁿ, we have (dx_i)_p(v_p) = v_i. This implies that (dx_i)_p((e_i)_p) = δ_ij for all i, j. Hence {(dx_i)_p : 1 ≤ i ≤ p} is the dual basis to {(ei)p: 1 ≤ i ≤ n}. We conclude that:

Theorem 2.2. The set {(dx_i)_p: 1 ≤ i ≤ n} forms a basis for T_p^∗Rⁿ. 3. Tangent Maps

Let U be an open subset of Rⁿ. Suppose F : U → R^m is a smooth map, where F = (F1, · · · , Fm). For each p ∈ U, we can define a linear map, called the tangent map,

dF_p : T_pRⁿ→ T_{F (p)}R^m as follows.

Let f be a germ at F (p). Then f ◦ F is a germ at p. Given any v_p∈ T_pRⁿ, we set (dFp(vp)) (f ) = vp(f ◦ F ).

Suppose F = F (y₁, · · · , y_m). By chain rule, ∂

∂x_i(p)(f ◦ F ) =

m

X

j=1

∂f

∂y_j(F (p))∂F_j

∂x_i(p). By definition, dFp

 ∂

∂xi

(p)

 (f ) =

m

X

j=1

∂F_j

∂xi

(p)∂f

∂yj

(F (p)). Hence we see that

dFp

 ∂

∂x_i(p)



=

m

X

j=1

∂Fj

∂x_i(p) ∂

∂y_j(F (p)).

If we choose the standard basis for TpRⁿand T_{F (p)}R^m, the tangent map dFp is represented by the Jacobi matrix of F at p.

Let U be an open set in Rⁿ and F : U → R^m be a smooth map. Assume that V is an open set in R^m containing F (U ) and G : V → R^kis a smooth map. For any x ∈ U, we have (chain rule)

(3.1) d(G ◦ F )x: TxRⁿ −−−−→ T^dFx _{F (x)}R^m

dG_{F (x)}

−−−−→ T_{G(F (x))}. 4. Differential Forms on Rⁿ

Let U be an open set in Rⁿ. We denote T U = [

p∈U

TpRⁿ, T^∗U = [

p∈U

T_p^∗Rⁿ.

Then we find that T U can be identify with U ×Rⁿand T^∗U can be identified with U ×(Rⁿ)^∗. We have a natural projection π : T^∗U → U.

Let f be a smooth function on an open set U ⊂ Rⁿ. The total derivative of f has the formal expression:

df =X

i

∂f

∂x_idx_i.

Since f is smooth, all the partial derivatives of f are all smooth. We shall use this idea to define the notion of one-form. A smooth one form ω on U is a formal expression

ω =X

i

ω_idx_i

with ωi ∈ C^∞(U ). Now we can think of ω as a map from U to T^∗U. In fact, for each p ∈ U, ω(p) =P

iωi(p)(dxi)p ∈ T_p^∗Rⁿ. Moreover, (π ◦ ω)(p) = p for all p ∈ U. Therefore a smooth one-form ω is a smooth map ω : U → T^∗U such that π ◦ ω = 1_U.

Let Λ^kT^∗U =S

p∈U Λ^kT_p^∗Rⁿ. We can also consider the natural projection¹ π : Λ^kT^∗U → U. We can check that Λ^kT^∗U can be identified with U × Λ^k(Rⁿ)^∗.

Definition 4.1. A smooth k-form on U is a smooth map η : U → Λ^kT^∗U such that π ◦ η = 1_U. The set of all smooth k forms on U is denoted by Ω^k(U ). We denote Ω^∗(U ) = L

k≥0Ω^k(U ).

By definition, a smooth k-form has the formal expression

η = X

1≤i1<···<ik≤n

η_i1···i_kdx_i1 ∧ · · · ∧ dx_ik with ηi1···ik ∈ C^∞(U ).

Let I = (i₁, · · · , i_k) be a k-tuple. We write

dxI = dxi1 ∧ · · · ∧ dx_ik. A k-form η is also denoted by η =P

Iη_Idx_I. The sum of two k-forms on U ω =P

Iω_Idx_I and η =P

IηIdxI is defined to be

ω + η =X

I

(ω + η)dx_I.

1Let us use the notation π for the projection but we have to know that this projection is different from the above.

For any smooth function f on U, we define f ω =X

I

(f ωI)dxI. One can check that

Proposition 4.1. The set Ω^k(U ) is a free C^∞(U )-module of rankn k

 .

On Ω^∗(U ), we define the exterior (wedge) product as follows. Given a s-form ω = P

Iω_Idx_I and a k-form η =P

Jη_Jdx_J, we define a s + k form by ω ∧ η =X

I,J

ω_Iη_Jdx_I∧ dx_J.

The exterior product of forms on U has the following properties.

Theorem 4.1. Let ω, η, and θ be k, s, r-forms respectively. Then (1) (ω ∧ η) ∧ θ = ω ∧ (η ∧ θ),

(2) ω ∧ η = (−1)^ksη ∧ ω,

(3) If r = s, then ω ∧ (η + θ) = ω ∧ η + ω ∧ θ.

Proof. Exercise. 

Let ω =P

Iω_Idx_I. We define the exterior derivative dω of ω by dω =X

I

dω_I∧ dx_I.

Proposition 4.2. d : Ω^∗(U ) → Ω^∗(U ) is a linear map such that (1) d : Ω^k(U ) → Ω^k+1(U )

(2) d(ω ∧ η) = dω ∧ η + (−1)^kω ∧ dη (3) d² = 0.

Proof. Exercise. 

Let dk= d|_Ω^k_{(U )}. Then d² = 0 implies that dk+1dk = 0. In other words, Im dk−1⊂ ker d_k. Definition 4.2. The k-th de Rham cohomology of U is the quotient space defined by

H^k(U ) = ker d_k/ Im d_k−1.

Later, we will study more about the de Rham cohomology of a smooth manifold.

4.1. Pull back on Differential Forms. Let F : U ⊂ Rⁿ→ R^m be a smooth map. Then F induces a linear map F^∗ from Ω^k(V ) to Ω^k(U ), where V is an open set containing F (U ).

Given a k-form ω on V, with k ≥ 1, we define a k-form F^∗ω on U by

(F^∗ω)(p)(v1, · · · , v_k) = ω(F (p))(dFp(v1), · · · , dFp(v_k)), ∀p ∈ U, where v₁, · · · , v_k ∈ T_pRⁿ. For k = 0, set F^∗g = g ◦ F.

Proposition 4.3. Let F : U ⊂ Rⁿ→ R^m and g ∈ C^∞(U ). Suppose ω, η are k-forms. Then (1) F^∗(ω + η) = F^∗ω + F^∗η,

(2) F^∗(gω) = (F^∗g)(F^∗ω),

(3) If ϕ₁, · · · , ϕ_k are one-forms, F^∗(ϕ₁∧ · · · ∧ ϕ_k) = F^∗ϕ₁∧ · · · ∧ F^∗ϕ_k.

Let us assume F = (F1, · · · , Fm), i.e. Fi = yi ◦ F. Suppose that ω = P

IωIdyI. Then F^∗ω =P

IF^∗ω_IF^∗dy_I. Using the properties, we have F^∗dy_i1∧· · ·∧F^∗dy_ik = dF_i1∧· · ·∧dF_ik. Then

F^∗ω =X

I

ω_I(F₁, · · · , F_m)dF_i1∧ · · · ∧ dF_ik. Using this identity, it is easy for us to prove the following corollary.

Corollary 4.1. Let ω, η be forms. Then F^∗(ω ∧ η) = F^∗ω ∧ F^∗η.

Proposition 4.4. Let F : U ⊂ Rⁿ→ R^m and G : V ⊂ R^m → R^k, where V is an open set containing F (U ). Then

(G ◦ F )^∗ω = F^∗(G^∗ω) for all k-forms ω.

Proposition 4.5. Let F : U → R^m be a smooth map and ω be a k-form. Then d(F^∗ω) = F^∗dω.