1 Dual Space Basics
1.1 Definition of the dual space V*
The dual space of a vector space V over a field F is the set of all linear functionals from V to F. A linear functional is a map that preserves vector addition and scalar multiplication. The dual space is commonly written as V*. It is itself a vector space under pointwise addition and scalar multiplication.
In practical terms, elements of V* assign a scalar value to each vector in V in a linear way. This makes the dual space a natural setting for studying vectors through measurements, projections, and other linear observations.
1.2 Linear functionals and the vector space structure
A linear functional f on V satisfies f(u + v) = f(u) + f(v) and f(cv) = cf(v) for all vectors u, v and scalars c. These conditions ensure that the set of all such maps is closed under addition and scalar multiplication defined by (f + g)(v) = f(v) + g(v) and (cf)(v) = c f(v).
Because these operations are defined pointwise, V* inherits the axioms of a vector space. The zero functional, which sends every vector to 0, serves as the additive identity.
1.3 Canonical pairing between V and V*
There is a natural bilinear pairing between V and V*, given by evaluation. If f is in V* and v is in V, then the pairing is written f(v). This pairing connects vectors with the functionals that test them.
The pairing is canonical because it requires no choice of basis or coordinates. It is a basic tool for expressing duality in linear algebra and for formulating more advanced constructions.
1.4 Dimension and existence of bases (finite-dimensional case)
If V is finite-dimensional and has dimension n, then V* also has dimension n. A basis of V determines a corresponding basis of V*, and vice versa. This is a fundamental result showing that finite-dimensional vector spaces and their duals are closely matched in size.
In finite dimensions, every linear functional is determined uniquely by its values on a basis of V. This makes dual spaces especially concrete and computable in ordinary linear algebra.
2 Bases, Coordinates, and the Dual Basis
2.1 Dual basis to a given basis of V
Suppose {v1, ..., vn} is a basis of V. The dual basis {φ1, ..., φn} in V* is defined by the rule φi(vj) = 1 when i = j and 0 when i ≠ j. Each dual basis vector picks out one coordinate relative to the chosen basis of V.
The dual basis is unique. It provides a systematic way to translate between vectors and their coordinate descriptions.
2.2 Coordinate functionals and matrix representation
Given a vector v = a1v1 + ... + anvn, the dual basis satisfies φi(v) = ai. Thus the dual basis functions as a family of coordinate functionals. Any linear functional f can be written as a linear combination f = b1φ1 + ... + bnφn.
In matrix language, a linear functional on a finite-dimensional space can be represented by a row vector once a basis is fixed. Evaluation then becomes ordinary matrix multiplication.
2.3 Change of basis and transformation rules
When the basis of V changes, the dual basis changes in the opposite way. If a change-of-basis matrix transforms vectors in V, then the corresponding transformation on V* is given by an inverse-transpose relationship. This reflects the contravariant nature of dualization.
As a result, coordinate descriptions of functionals depend on the chosen basis, but the underlying linear map remains the same. The dual space captures the same functional information in a different coordinate system.
2.4 Identifying V* with coordinate spaces
After choosing a basis, V* can be identified with the space of row vectors or coordinate tuples of scalars. Under this identification, each functional corresponds to a list of coefficients acting on column vectors.
This identification is not canonical, since it depends on the selected basis. It is nonetheless very useful for explicit computation and for connecting abstract duality with familiar coordinate algebra.
3 Duals of Linear Maps
3.1 The dual (transpose) map of a linear transformation
If T: V → W is linear, its dual map T*: W* → V* is defined by T*(λ) = λ ∘ T. In words, a functional on W is pulled back along T to produce a functional on V. This construction is also called the transpose or adjoint in purely algebraic settings.
The map T* is linear and reverses the direction of arrows. It is one of the central mechanisms by which dual spaces interact with linear maps.
3.2 Functorial properties of dualization
Dualization is contravariant: it turns maps V → W into maps W* → V*. This reversal is systematic and behaves consistently across compositions and identities. In categorical language, it defines a contravariant functor from vector spaces to vector spaces.
This functorial behavior explains why dual spaces are so widely used in structural mathematics. They allow properties of maps to be studied from the dual perspective without losing linearity.
3.3 Kernel and image relationships under dual maps
The kernel of T* consists of those functionals on W that vanish on the image of T. Thus ker(T*) is the annihilator of im(T). Likewise, the image of T* is related to functionals on V that factor through T.
In finite dimensions, these relationships lead to familiar dimension formulas. In infinite-dimensional settings, the same inclusions hold, but surjectivity and injectivity properties may differ sharply from the finite-dimensional case.
3.4 Composition and identity compatibility
For linear maps S: U → V and T: V → W, the dual of the composition satisfies (T ∘ S)* = S* ∘ T*. Also, the dual of the identity map is again the identity map. These properties confirm that dualization respects the basic algebraic structure of linear maps.
Together, they make the dual construction reliable for building more elaborate arguments. The order reversal in composition is a hallmark of contravariant processes.
4 Double Dual and Natural Maps
4.1 The evaluation map V → V**
There is a canonical map from V to its double dual V**, defined by sending each vector v to the functional evv on V* given by evv(f) = f(v). This is called the evaluation map or natural embedding.
It uses the basic pairing between vectors and functionals to produce a second-level functional. The construction requires no additional choices.
4.2 Properties of the evaluation map
The evaluation map is always linear. It is injective for every vector space, because a nonzero vector can be separated from zero by some linear functional in standard vector space settings. Thus V may be regarded as a subspace of V**.
This embedding preserves much of the structure of V. It is a fundamental bridge between a space and the functionals defined on its dual.
4.3 Conditions for natural isomorphism (finite-dimensional case)
When V is finite-dimensional, the evaluation map V → V** is an isomorphism. In this case, every functional on V* arises from evaluation at a unique vector in V. The identification is canonical, meaning it does not depend on a chosen basis.
This natural isomorphism is one of the most important finite-dimensional duality results. It shows that finite-dimensional spaces and their double duals are essentially the same object.
4.4 Interpreting V** and reflexivity
The double dual V** consists of all linear functionals on V*. It can be viewed as a space of higher-order linear observations on V. A vector space is called reflexive, in the basic algebraic sense, when the canonical map into its double dual is an isomorphism.
All finite-dimensional vector spaces are reflexive. In infinite dimensions, reflexivity may fail, and the double dual can be strictly larger than the original space.
5 Annihilators and Subspace Duality
5.1 Annihilators of subspaces
If U is a subspace of V, its annihilator U^0 in V* is the set of all linear functionals that vanish on U. This is a subspace of V* and records exactly which functionals cannot detect vectors from U.
Annihilators provide a precise way to translate subspace information into the dual setting. They are essential in understanding how subspaces and quotient spaces interact under dualization.
5.2 Relationship between annihilators and quotients
A quotient V/U gives rise to a natural identification of (V/U)* with U^0. A functional on V/U corresponds exactly to a functional on V that is zero on U. This is one of the cleanest links between dual spaces and quotient constructions.
The correspondence is canonical and works in all dimensions. It is widely used to transfer problems about quotient spaces into problems about subspaces of the dual.
5.3 Orthogonality-type correspondences (purely algebraic)
Annihilators behave like algebraic analogues of orthogonal complements. Instead of a dot product, the pairing between V and V* determines which vectors are “invisible” to a given family of functionals. This creates a duality between subspaces of V and subspaces of V*.
Unlike geometric orthogonality, this correspondence depends only on linear structure. It is therefore applicable in settings where no inner product is available.
5.4 Lattice relationships among subspaces and annihilators
Annihilators reverse inclusion: if U ⊆ W, then W^0 ⊆ U^0. They also satisfy identities involving sums and intersections, such as the annihilator of a sum being the intersection of annihilators. These relations make annihilators compatible with the lattice structure of subspaces.
In finite dimensions, double annihilators recover the original subspaces. This gives a precise duality between the subspace lattice of V and that of V*.
6 Tensor-Product Viewpoint (Optional Algebraic Extensions)
6.1 Tensor products and multilinear functionals
Tensor products provide a framework for converting multilinear maps into linear ones. A bilinear map on V × W can be represented as a linear map on V ⊗ W. This is especially useful for organizing identities involving dual spaces.
From this perspective, duality is closely tied to how linear functionals extend across tensor constructions. The tensor product encodes the universal behavior of multilinear expressions.
6.2 Duals as linear maps out of tensors
Linear functionals on a vector space can be interpreted through tensorial language when the space is built from generators and relations. More generally, a bilinear pairing V × V* → F is a manifestation of the universal role of linear maps into the base field.
This viewpoint helps unify the treatment of functionals, pairings, and multilinear forms. It also clarifies why dual spaces appear naturally in tensor calculus.
6.3 Hom–tensor adjunction (algebraic formulation)
A standard algebraic principle states that linear maps from a tensor product correspond to bilinear maps from the factors. In one form, Hom(V ⊗ W, F) is naturally identified with the space of bilinear forms on V × W. Such identifications are central in abstract linear algebra.
These adjunctions express the fact that tensor products and duals are complementary constructions. One packages multilinearity; the other packages linear testing.
6.4 Connecting dual space constructions to tensors
Dual spaces can be incorporated into tensor products through natural pairings and contraction operations. For example, an element of V* can be paired with an element of V to produce a scalar, and this can be extended across more elaborate tensor expressions.
This interaction underlies many algebraic formulas involving covariant and contravariant behavior. It also provides a bridge to multilinear algebra and representation theory.
7 Examples and Computations
7.1 Dual of R^n and explicit coordinate functionals
For R^n, every linear functional has the form f(x1, ..., xn) = a1x1 + ... + anxn for uniquely determined scalars a1, ..., an. Hence (R^n)* is naturally identified with R^n after choosing the standard basis. The coordinate functionals are simply the projections onto each coordinate.
This example is the model case for dual spaces. It shows how abstract definitions reduce to familiar linear expressions.
7.2 Duals of spaces of polynomials (finite degree)
Let Pn be the vector space of polynomials of degree at most n. Its dual consists of all linear functionals determined by values on a basis such as 1, x, x^2, ..., x^n. For instance, evaluation at a point a, sending p(x) to p(a), is a linear functional on Pn.
Other examples include coefficient extraction and differentiation at a point. These illustrate how dual spaces arise naturally in algebraic contexts beyond coordinate vectors.
7.3 Duals of function spaces with algebraic linear structure
Many spaces of functions, such as spaces of sequences or polynomial functions, can be treated as vector spaces over a field. Their duals contain linear maps that may be defined by finite linear combinations of values, coefficients, or other algebraic data.
In infinite-dimensional settings, the dual may be very large and difficult to describe completely. Still, the defining principle remains the same: elements of the dual are linear observations on the original space.
7.4 Worked example: constructing bases and dual bases
Consider a two-dimensional vector space with basis {v1, v2}. The dual basis {φ1, φ2} is determined by φ1(v1) = 1, φ1(v2) = 0 and φ2(v1) = 0, φ2(v2) = 1. Any vector v = av1 + bv2 then satisfies φ1(v) = a and φ2(v) = b.
Any linear functional f is therefore of the form f = aφ1 + bφ2 for some scalars a and b. This concrete calculation demonstrates how bases and dual bases encode coordinates and linear measurements.
8 Common Identities and Theorems
8.1 Rank-nullity implications for dual maps
For a linear map T: V → W between finite-dimensional spaces, rank-nullity relates the dimensions of ker(T) and im(T). Applying dualization yields complementary relations for T*. The dimensions of the kernel and image of the dual reflect those of the original map.
These formulas are often used to compare injectivity and surjectivity between a map and its dual. In finite dimensions, T is injective precisely when T* is surjective, and T is surjective precisely when T* is injective.
8.2 Dimension formulas involving annihilators
If V is finite-dimensional and U is a subspace, then dim(U) + dim(U^0) = dim(V). This formula shows that annihilators measure the complementary size of subspaces in the dual setting.
Similarly, the annihilator of the annihilator, U^00, corresponds back to U under the natural identification of V with V**. These formulas are among the most useful computational tools in duality theory.
8.3 Behavior of subspaces under dualization
Subspaces transform under duality in a reversed manner. Inclusion relations reverse, sums become intersections after taking annihilators, and quotient structures correspond to subspaces of the dual. This pattern is consistent across many standard results.
Such behavior makes dualization a powerful method for translating algebraic problems into equivalent dual statements. Often, a theorem about subspaces has a companion theorem about annihilators.
8.4 Canonical identifications and their limits (infinite-dimensional)
Several identifications in duality are canonical only in finite dimensions. In particular, V and V** need not be isomorphic when V is infinite-dimensional, and V* may be much larger or behave differently from coordinate space models.
This limitation is important because it marks the boundary between elementary linear algebra and more advanced functional-analytic settings. Even so, the basic constructions of dual spaces, dual maps, and annihilators remain valid in full generality.