15 The Dual Space
This chapter is more demanding than the surrounding chapters and assumes familiarity with abstract algebra, particularly groups, quotient groups, and the first isomorphism theorem. The connection between annihilators and quotient spaces in Section 15.2 uses the linear algebra analogue of G/\ker\phi \cong \operatorname{im}\phi — students who have seen the group version will find the argument immediate.
The double dual in Section 15.3 requires careful attention to what depends on choices versus what does not. This is made precise without any machinery beyond what appears in this course, but it demands a degree of abstraction not required in earlier chapters.
Students not yet comfortable with abstract algebra can treat this chapter as optional enrichment. No later chapter depends on it.
In Chapter 3 we introduced the dual space \mathcal{V}^* = \mathcal{L}(\mathcal{V}, \mathbb{R}), the dual basis, and the transpose T^*. The main results there were that \dim \mathcal{V}^* = \dim \mathcal{V} and that T^* reverses arrows contravariantly. Those facts were developed as tools for studying linear maps and their matrix representations.
This chapter returns to the dual space as an object worthy of study in its own right. The questions we ask are different: not “how do we compute with functionals?” but “what is the relationship between \mathcal{V} and \mathcal{V}^*, and why is the relationship between \mathcal{V} and \mathcal{V}^{**} fundamentally different?”
The central theme is the distinction between identifications that depend on choices — a basis, an inner product — and identifications that do not. This distinction, invisible in elementary linear algebra, becomes fundamental in geometry, analysis, and algebra.
15.1 The Natural Pairing
We begin by organizing the relationship between \mathcal{V} and \mathcal{V}^* in a way that emphasizes its symmetry.
The natural pairing between \mathcal{V}^* and \mathcal{V} is the evaluation map \langle \varphi, v \rangle = \varphi(v), \quad \varphi \in \mathcal{V}^*, \; v \in \mathcal{V}.
The bracket notation makes the symmetry of the relationship visible: a functional evaluates a vector, and we can think of either as the “active” participant. The pairing is bilinear — linear in \varphi (since \mathcal{V}^* is a vector space) and linear in v (since each \varphi is linear): \langle a\varphi + b\psi, v \rangle = a\langle\varphi, v\rangle + b\langle\psi, v\rangle, \qquad \langle \varphi, au + bv \rangle = a\langle\varphi, u\rangle + b\langle\varphi, v\rangle.
An inner product on \mathcal{V} requires both arguments from the same space. The natural pairing takes \varphi \in \mathcal{V}^* and v \in \mathcal{V} from two different spaces, so it is not an inner product. An inner product on \mathcal{V} does induce an identification \mathcal{V} \cong \mathcal{V}^* by v \mapsto \langle v, \cdot \rangle, but this requires extra structure. The natural pairing requires none.
The pairing is non-degenerate in both arguments: if \langle \varphi, v \rangle = 0 for all v, then \varphi = 0; and if \langle \varphi, v \rangle = 0 for all \varphi, then v = 0. The second statement follows because for any nonzero v we can extend it to a basis, take the corresponding dual basis functional \varphi_1, and get \langle \varphi_1, v \rangle = 1 \neq 0.
Non-degeneracy means \mathcal{V} and \mathcal{V}^* “see” each other completely through the pairing — no vector is invisible to all functionals, and no functional is invisible to all vectors. This is the sense in which \mathcal{V} and \mathcal{V}^* form a dual pair.
15.1.1 What basis-dependence looks like
Recall from Chapter 3 that the dual basis \mathcal{B}^* = \{\varphi_1, \ldots, \varphi_n\} is determined by \varphi_i(v_j) = \delta_{ij}, and gives an isomorphism \mathcal{V} \to \mathcal{V}^* sending v_i \mapsto \varphi_i. But this isomorphism changes with the basis. To see how, suppose \mathcal{B} = \{v_1, \ldots, v_n\} and \mathcal{C} = \{w_1, \ldots, w_n\} are two bases, related by w_j = \sum_i P_{ij} v_i (so the columns of P express the new basis in the old).
Theorem 15.1 If P is the change-of-basis matrix from \mathcal{B} to \mathcal{C}, then for any \phi \in \mathcal{V}^*, [\phi]_{\mathcal{B}^*} = (P^{-1})^T [\phi]_{\mathcal{C}^*}.
Proof. Let d_k = \phi(v_k) and e_k = \phi(w_k) be the \mathcal{B}^* and \mathcal{C}^* coordinates respectively. Since w_j = \sum_i P_{ij} v_i: e_j = \phi(w_j) = \sum_i P_{ij} \phi(v_i) = (P^T d)_j. So e = P^T d, giving d = (P^T)^{-1} e = (P^{-1})^T e. \square
Vectors in \mathcal{V} transform by P^{-1} when passing from \mathcal{B}-coordinates to \mathcal{C}-coordinates (since [v]_\mathcal{C} = P^{-1}[v]_\mathcal{B}). Functionals transform by (P^{-1})^T. These are different transformations unless P is orthogonal. This is contravariance in coordinates: the dual basis transforms by the contragredient of the change-of-basis matrix, not by the matrix itself.
15.2 Annihilators
Given a subspace \mathcal{W} \subseteq \mathcal{V}, there is a natural subspace of \mathcal{V}^* associated to it: all functionals that vanish identically on \mathcal{W}.
Definition 15.1 (Annihilator) The annihilator of \mathcal{W} \subseteq \mathcal{V} is \mathcal{W}^0 = \{ \varphi \in \mathcal{V}^* : \langle \varphi, w \rangle = 0 \text{ for all } w \in \mathcal{W} \}.
The annihilator is a subspace of \mathcal{V}^*: it contains the zero functional, is closed under addition (if \varphi, \psi both vanish on \mathcal{W}, so does \varphi + \psi), and closed under scalar multiplication. A small subspace \mathcal{W} imposes few vanishing conditions, so \mathcal{W}^0 is large; a large \mathcal{W} imposes many conditions, so \mathcal{W}^0 is small.
Theorem 15.2 If \mathcal{V} is finite-dimensional and \mathcal{W} \subseteq \mathcal{V}, then \dim \mathcal{W}^0 = \dim \mathcal{V} - \dim \mathcal{W}.
Proof. Let \dim \mathcal{V} = n and \dim \mathcal{W} = k. Choose a basis \{w_1, \ldots, w_k\} of \mathcal{W} and extend to a basis \{w_1, \ldots, w_k, v_{k+1}, \ldots, v_n\} of \mathcal{V}. Let \{\varphi_1, \ldots, \varphi_k, \psi_{k+1}, \ldots, \psi_n\} be the corresponding dual basis.
We claim \mathcal{W}^0 = \operatorname{span}\{\psi_{k+1}, \ldots, \psi_n\}.
Each \psi_j with j > k satisfies \psi_j(w_\ell) = \delta_{j\ell} = 0 for \ell \le k, so it vanishes on all of \mathcal{W}. Thus \operatorname{span}\{\psi_{k+1}, \ldots, \psi_n\} \subseteq \mathcal{W}^0.
Conversely, if \phi \in \mathcal{W}^0, write \phi = \sum_{i=1}^k a_i \varphi_i + \sum_{j=k+1}^n b_j \psi_j and evaluate on w_\ell for \ell \le k: 0 = \phi(w_\ell) = a_\ell, since \psi_j(w_\ell) = 0 for j > k and \varphi_i(w_\ell) = \delta_{i\ell}. So all a_i = 0, meaning \phi \in \operatorname{span}\{\psi_{k+1}, \ldots, \psi_n\}. \square
Compare with the orthogonal complement formula \dim \mathcal{W}^\perp = \dim \mathcal{V} - \dim \mathcal{W} from the Orthogonality chapter. The annihilator gives the same count without any inner product — but at the cost of \mathcal{W}^0 living in the different space \mathcal{V}^* rather than back inside \mathcal{V}.
15.2.1 Annihilators and the transpose
The annihilator organizes the kernel and image of the transpose cleanly.
Theorem 15.3 For T : \mathcal{V} \to \mathcal{W} linear: \ker(T^*) = (\operatorname{im} T)^0, \qquad \operatorname{im}(T^*) = (\ker T)^0.
Proof. For the first: \varphi \in \ker(T^*) iff (T^*\varphi)(v) = \varphi(T(v)) = 0 for all v iff \varphi vanishes on \operatorname{im}(T) iff \varphi \in (\operatorname{im} T)^0.
For the second: if \varphi = T^*\psi and v \in \ker T, then \varphi(v) = \psi(T(v)) = \psi(0) = 0, so \operatorname{im}(T^*) \subseteq (\ker T)^0. Both sides have dimension \dim \mathcal{V} - \dim \ker T by rank–nullity and Theorem 15.2, so equality holds. \square
The first identity gives a cleaner proof that \dim \operatorname{im}(T^*) = \dim \operatorname{im}(T) than the one in Chapter 3: rank–nullity for T^* gives \dim \operatorname{im}(T^*) = \dim \mathcal{W} - \dim\ker(T^*) = \dim \mathcal{W} - (\dim \mathcal{W} - \dim \operatorname{im} T) = \dim \operatorname{im} T.
15.2.2 Connection to quotient spaces
The following theorem uses quotient vector spaces and the first isomorphism theorem. The quotient \mathcal{V}/\mathcal{W} is the set of cosets \{v + \mathcal{W} : v \in \mathcal{V}\} with the induced linear structure. The first isomorphism theorem for vector spaces states: for any linear map T : \mathcal{V} \to \mathcal{U}, there is an isomorphism \mathcal{V}/\ker(T) \xrightarrow{\sim} \operatorname{im}(T) given by v + \ker(T) \mapsto T(v). This is the direct analogue of G/\ker\phi \cong \operatorname{im}\phi for groups — the same proof works in both settings.
Theorem 15.4 There is an isomorphism \mathcal{W}^0 \cong (\mathcal{V}/\mathcal{W})^*.
Proof. Define \Phi : \mathcal{W}^0 \to (\mathcal{V}/\mathcal{W})^* by \Phi(\varphi)(v + \mathcal{W}) = \varphi(v). This is well-defined: if v - v' \in \mathcal{W} then \varphi(v - v') = 0 since \varphi \in \mathcal{W}^0. Each \Phi(\varphi) is linear since \varphi is linear.
\Phi is linear. \Phi is injective: if \Phi(\varphi) = 0 then \varphi = 0. \Phi is surjective: any \bar\psi \in (\mathcal{V}/\mathcal{W})^* pulls back via the quotient map \pi : \mathcal{V} \to \mathcal{V}/\mathcal{W} to \psi = \bar\psi \circ \pi \in \mathcal{V}^*, and \psi \in \mathcal{W}^0 since \psi(w) = \bar\psi(w + \mathcal{W}) = \bar\psi(0 + \mathcal{W}) = 0 for w \in \mathcal{W}. \square
The analogy with groups is exact: homomorphisms G \to H that are trivial on N correspond to homomorphisms G/N \to H. Functionals on \mathcal{V} that vanish on \mathcal{W} correspond to functionals on \mathcal{V}/\mathcal{W}.
15.3 The Double Dual
Applying the dual space construction twice gives the double dual \mathcal{V}^{**} = (\mathcal{V}^*)^*. By the dimension theorem from Chapter 3, \dim \mathcal{V}^{**} = \dim \mathcal{V} in finite dimensions, so \mathcal{V} and \mathcal{V}^{**} are isomorphic. The point of this section is that there is a specific isomorphism requiring no choices at all.
Definition 15.2 (Evaluation map) Define \iota : \mathcal{V} \to \mathcal{V}^{**} by \iota(v)(\varphi) = \varphi(v) = \langle \varphi, v \rangle, \quad v \in \mathcal{V}, \; \varphi \in \mathcal{V}^*. That is, \iota(v) is the element of \mathcal{V}^{**} that evaluates a functional at v.
Each vector v \in \mathcal{V} gives rise to a functional on \mathcal{V}^* by “evaluate at v.” No basis is chosen; no inner product is used.
Theorem 15.5 The map \iota : \mathcal{V} \to \mathcal{V}^{**} is always injective. If \mathcal{V} is finite-dimensional, it is an isomorphism.
Proof. Linearity: \iota(au + bv)(\varphi) = \varphi(au + bv) = a\varphi(u) + b\varphi(v) = (a\iota(u) + b\iota(v))(\varphi) for all \varphi.
Injectivity: if \iota(v) = 0, then \varphi(v) = 0 for all \varphi \in \mathcal{V}^*. By non-degeneracy of the natural pairing, v = 0.
In finite dimensions, an injective map between spaces of equal dimension is an isomorphism. \square
15.3.1 What “canonical” means here
Since \dim \mathcal{V} = \dim \mathcal{V}^*, there are many isomorphisms \mathcal{V} \to \mathcal{V}^* — a different one for each basis. The isomorphism \iota : \mathcal{V} \to \mathcal{V}^{**} is different in character: it is defined entirely from the evaluation pairing with no choices made anywhere. We say it is canonical.
The content of this claim is that \iota commutes with every linear map simultaneously.
Theorem 15.6 For any linear map T : \mathcal{V} \to \mathcal{W}, the following diagram commutes: \begin{array}{ccc} \mathcal{V} & \xrightarrow{\;\iota_{\mathcal{V}}\;} & \mathcal{V}^{**} \\[6pt] \downarrow\scriptstyle{T} & & \downarrow\scriptstyle{T^{**}} \\[6pt] \mathcal{W} & \xrightarrow{\;\iota_{\mathcal{W}}\;} & \mathcal{W}^{**} \end{array} That is, T^{**} \circ \iota_{\mathcal{V}} = \iota_{\mathcal{W}} \circ T, where T^{**} = (T^*)^*.
Proof. For v \in \mathcal{V} and \psi \in \mathcal{W}^*: (T^{**} \circ \iota_{\mathcal{V}})(v)(\psi) = \iota_{\mathcal{V}}(v)(T^*\psi) = (T^*\psi)(v) = \psi(T(v)), (\iota_{\mathcal{W}} \circ T)(v)(\psi) = \iota_{\mathcal{W}}(T(v))(\psi) = \psi(T(v)). These are equal. \square
The commutativity of this diagram for every T simultaneously is precisely what canonicity means concretely. The basis-dependent isomorphism \mathcal{V} \cong \mathcal{V}^* fails this test: if T : \mathcal{V} \to \mathcal{V} does not preserve the basis, the analogous diagram does not commute.
15.3.2 Why \mathcal{V} \not\cong \mathcal{V}^* canonically
The argument is a simple obstruction. If a canonical isomorphism \iota' : \mathcal{V} \to \mathcal{V}^* existed, it would have to commute with every invertible T : \mathcal{V} \to \mathcal{V}: T^* \circ \iota' = \iota' \circ T. In coordinates relative to any fixed basis, T acts by a matrix P and T^* acts by (P^{-1})^T. For the equation above to hold for all invertible P, the isomorphism \iota' would need to simultaneously intertwine P and (P^{-1})^T for every invertible matrix. Taking P = 2I (scalar scaling), we would need (2I)^{-T} = (2I), i.e., \frac{1}{2} = 2. No such isomorphism exists.
An inner product breaks this obstruction by providing an identification v \leftrightarrow \langle v, \cdot \rangle that is canonical given the inner product — but it depends on which inner product is chosen, and changes when the inner product changes.
15.3.3 The infinite-dimensional case
In infinite dimensions, \iota : \mathcal{V} \to \mathcal{V}^{**} is always injective but typically fails to be surjective: \mathcal{V}^{**} is strictly larger than \mathcal{V}. For \mathcal{V} = \ell^1(\mathbb{N}) one has \mathcal{V}^* \cong \ell^\infty(\mathbb{N}) and \mathcal{V}^{**} \supsetneq \mathcal{V}. Spaces for which \iota is surjective are called reflexive and include all finite-dimensional spaces, all Hilbert spaces, and the L^p spaces for 1 < p < \infty.
15.4 Closing Remarks
Chapter 3 introduced the dual space as a computational tool. This chapter has developed it as a structure. The natural pairing between \mathcal{V} and \mathcal{V}^* requires no choices and makes both spaces visible to each other. The contravariant transformation law (Theorem 15.1) shows precisely how the basis-dependent isomorphism \mathcal{V} \cong \mathcal{V}^* fails to be natural. The annihilator \mathcal{W}^0 \subseteq \mathcal{V}^* mirrors the orthogonal complement dimensionally — without needing an inner product — and the isomorphism \mathcal{W}^0 \cong (\mathcal{V}/\mathcal{W})^* connects the dual to quotient spaces exactly as the first isomorphism theorem connects kernels to quotients for group homomorphisms.
The double dual resolves the asymmetry: \iota : \mathcal{V} \xrightarrow{\sim} \mathcal{V}^{**} is canonical, commuting with all linear maps simultaneously. This is the first instance in the course where the difference between “there exists an isomorphism” and “there exists a canonical isomorphism” becomes mathematically essential rather than philosophically interesting. The language developed here — natural pairing, annihilators, contravariance, reflexivity — reappears in differential geometry (tangent and cotangent bundles), algebraic topology (cohomology), and representation theory (contragredient representations).