# Dual space
In linear algebra, the **dual space** of a vector space \( V \) over a field \( \mathbb{F} \) is the vector space \( V^{*} \) consisting of all linear functionals on \( V \), that is, all linear maps from \( V \) to the scalar field \( \mathbb{F} \)[1](https://sites.math.northwestern.edu/scg479/courses/notes/dual-spaces.pdf). Linear functionals can be added and multiplied by scalars pointwise, \( (\varphi + \psi)(\mathbf{v}) = \varphi(\mathbf{v}) + \psi(\mathbf{v}) \) and \( (\lambda\varphi)(\mathbf{v}) = \lambda\,\varphi(\mathbf{v}) \), and under these operations \( V^{*} \) is itself a vector space[2](https://www.maths.tcd.ie/~dwilkins/Courses/MAU34802/MAU34802_Hil2021_CourseNotes/MAU34802_Hil2021_Section2.pdf). Equivalently, \( V^{*} = \mathcal{L}(V, \mathbb{F}) \), the space of linear maps from \( V \) to \( \mathbb{F} \)[1](https://sites.math.northwestern.edu/scg479/courses/notes/dual-spaces.pdf).
| Key fact | Detail |
|---|---|
| Definition | \( V^{*} \) is the space of all linear maps \( V \to \mathbb{F} \); it is a vector space under pointwise addition and scalar multiplication[1](https://sites.math.northwestern.edu/scg479/courses/notes/dual-spaces.pdf) • [2](https://www.maths.tcd.ie/~dwilkins/Courses/MAU34802/MAU34802_Hil2021_CourseNotes/MAU34802_Hil2021_Section2.pdf) |
| Dimension | If \( V \) is finite-dimensional with basis \( (v_1, \ldots, v_n) \), the dual basis \( (f_1, \ldots, f_n) \) with \( f_i(v_j) = 1 \) for \( i = j \) and \( 0 \) otherwise is a basis of \( V^{*} \), so \( \dim V^{*} = \dim V \)[1](https://sites.math.northwestern.edu/scg479/courses/notes/dual-spaces.pdf) |
| No canonical self-identification | There is in general no canonical isomorphism between \( V \) and \( V^{*} \), even when \( V \) is finite-dimensional[3](https://people.math.harvard.edu/~elkies/M55a.02/dual.html) |
| Double dual | Each \( v \in V \) acts on \( V^{*} \) by evaluation, giving a canonical map \( V \to V^{**} \) that is an isomorphism when \( V \) is finite-dimensional[3](https://people.math.harvard.edu/~elkies/M55a.02/dual.html) |
| Annihilator | For a subspace \( U \subset V \), the annihilator \( U^{\perp} \subset V^{*} \) of functionals vanishing on \( U \) is naturally isomorphic to \( (V/U)^{*} \), and \( \dim U^{\perp} \) equals the codimension of \( U \)[1](https://sites.math.northwestern.edu/scg479/courses/notes/dual-spaces.pdf) • [3](https://people.math.harvard.edu/~elkies/M55a.02/dual.html) |
| Topological dual | For a normed space \( X \), the topological dual \( X^{*} \) is the subspace of bounded (equivalently, continuous) linear functionals with the operator norm; it is always a Banach space, and in infinite dimensions \( X^{*} \) is a proper subspace of the algebraic dual[4](https://www.buttenschoen.ca/MATH725/duality/duality/) |
## Linear functionals and examples
A linear functional is a linear transformation \( \varphi: V \to \mathbb{F} \) from a vector space to its field of scalars[5](https://people.tamu.edu/~f-narcowich/m641/m641_notes/vector_spaces.html). When \( V = \mathbb{R}^n \), every linear functional has the form \( f(x_1, \ldots, x_n) = \xi_1 x_1 + \cdots + \xi_n x_n \) for fixed coefficients \( \xi_1, \ldots, \xi_n \), so functionals on \( \mathbb{R}^n \) can be written as row vectors acting on column vectors[6](https://people.math.osu.edu/gerlach.1/math5101/5101lecturenotes/DualOfAVectorSpace.pdf). If a basis of \( V \) is chosen and vectors are identified with column vectors of length \( n \), elements of \( V^{*} \) correspond to row vectors of the same length[3](https://people.math.harvard.edu/~elkies/M55a.02/dual.html).
Functionals also arise on function spaces. On \( V = C[a,b] \), the vector space of functions continuous on a closed interval \( [a,b] \), three standard examples are the point evaluation \( \psi \mapsto \psi(s_1) \) at a fixed point \( s_1 \), a weighted sum of samples \( \psi \mapsto \sum_{j=1}^{n} k_j \psi(s_j) \) at chosen points, and the integral \( \psi \mapsto \int_a^b \psi(s)\,ds \); each is linear[6](https://people.math.osu.edu/gerlach.1/math5101/5101lecturenotes/DualOfAVectorSpace.pdf).
## The dual basis and dimension
Suppose \( V \) is finite-dimensional with basis \( (v_1, \ldots, v_n) \). For each \( i \), define \( f_i: V \to \mathbb{F} \) by \( f_i(v_j) = 1 \) if \( i = j \) and \( 0 \) otherwise, extended linearly to all of \( V \). The functionals \( (f_1, \ldots, f_n) \) form a basis of \( V^{*} \), called the **dual basis** of \( (v_1, \ldots, v_n) \)[1](https://sites.math.northwestern.edu/scg479/courses/notes/dual-spaces.pdf). In the notation of Wilkins's Trinity College lecture notes, the coordinate functionals \( \varepsilon_i \) extract the \( i \)-th coefficient: \( \varepsilon_i(\sum_{j=1}^{n} \lambda_j u_j) = \lambda_i \), and every \( \varphi \in V^{*} \) expands as \( \varphi = \sum_{i=1}^{n} \varphi(u_i)\,\varepsilon_i \)[2](https://www.maths.tcd.ie/~dwilkins/Courses/MAU34802/MAU34802_Hil2021_CourseNotes/MAU34802_Hil2021_Section2.pdf).
It follows that \( \dim V^{*} = \dim V \), so a finite-dimensional space and its dual are isomorphic[1](https://sites.math.northwestern.edu/scg479/courses/notes/dual-spaces.pdf). The isomorphism, however, depends on the choice of basis: there is no canonical isomorphism between \( V \) and \( V^{*} \)[3](https://people.math.harvard.edu/~elkies/M55a.02/dual.html). In infinite dimensions the [dual basis](https://www.edgepedia.org/dual-basis) construction breaks down: the coordinate functionals remain linearly independent but in general do not span the whole dual space[7](https://danielmiranda.prof.ufabc.edu.br/livros/ala/capitulo-05-espacos-duais.html).
## The double dual
Since \( V^{*} \) is itself a vector space, it has a dual \( V^{**} \), the **double dual** of \( V \)[2](https://www.maths.tcd.ie/~dwilkins/Courses/MAU34802/MAU34802_Hil2021_CourseNotes/MAU34802_Hil2021_Section2.pdf). The pairing \( \varphi(\mathbf{v}) \) is bilinear: it is linear in \( \mathbf{v} \) for each fixed \( \varphi \), and linear in \( \varphi \) for each fixed \( \mathbf{v} \). Each vector \( \mathbf{v} \) therefore defines a functional on \( V^{*} \) by evaluation, \( E_v(\varphi) = \varphi(v) \), which is an element of \( V^{**} \)[2](https://www.maths.tcd.ie/~dwilkins/Courses/MAU34802/MAU34802_Hil2021_CourseNotes/MAU34802_Hil2021_Section2.pdf) • [3](https://people.math.harvard.edu/~elkies/M55a.02/dual.html).
This evaluation map \( V \to V^{**} \) is defined without reference to any basis, and it is an isomorphism when \( V \) is finite-dimensional[3](https://people.math.harvard.edu/~elkies/M55a.02/dual.html). This is the structural reason the double dual is better behaved than the single dual: \( V \) is *naturally* isomorphic to \( V^{**} \), while the isomorphism \( V \cong V^{*} \) requires a choice of basis[3](https://people.math.harvard.edu/~elkies/M55a.02/dual.html).
## Induced maps and annihilators
A linear transformation \( T: V \to W \) induces a map \( T^{*}: W^{*} \to V^{*} \), called the adjoint (or dual, or transpose), defined by \( T^{*}(f) = f \circ T \) for \( f \in W^{*} \); the assignment \( T \mapsto T^{*} \) is itself a linear map, and it is an isomorphism between \( \mathcal{L}(V,W) \) and \( \mathcal{L}(W^{*},V^{*}) \) when \( V \) and \( W \) are finite-dimensional[3](https://people.math.harvard.edu/~elkies/M55a.02/dual.html) • [8](https://math.virginia.edu/graduate/exams/algebra/alg-prep-4-1.html).
For a subspace \( U \) of \( V \), the **annihilator** \( U^{\perp} \) is the set of linear functionals on \( V \) that vanish on \( U \); it is a subspace of \( V^{*} \)[1](https://sites.math.northwestern.edu/scg479/courses/notes/dual-spaces.pdf). There is a natural isomorphism between \( U^{\perp} \) and the dual \( (V/U)^{*} \) of the quotient space[1](https://sites.math.northwestern.edu/scg479/courses/notes/dual-spaces.pdf). For finite-dimensional \( V \), the dimension of \( U^{\perp} \) is the codimension of \( U \) within \( V \); moreover, when \( W \) decomposes as the direct sum of \( U \) and \( V \), its dual \( W^{*} \) likewise splits into \( U^{*} \) and \( V^{*} \), with the \( U^{*} \) summand coinciding with the annihilator of \( U \)[3](https://people.math.harvard.edu/~elkies/M55a.02/dual.html). When \( V \) carries an inner product, the annihilator construction corresponds to taking the orthogonal complement of a subspace, which is why dual-space notions can rephrase inner-product statements without using an inner product[1](https://sites.math.northwestern.edu/scg479/courses/notes/dual-spaces.pdf).
## Algebraic and topological duals
The definition above gives the **algebraic dual**: all linear functionals, with no continuity requirement[4](https://www.buttenschoen.ca/MATH725/duality/duality/). When \( X = (V, \|\cdot\|) \) is a normed space, the **topological dual** \( X^{*} \) is the subspace of bounded, equivalently continuous, linear functionals \( f: V \to \mathbb{R} \), equipped with the operator norm \( \|f\|_{X^{*}} = \sup_{\|x\| \le 1} |f(x)| \)[4](https://www.buttenschoen.ca/MATH725/duality/duality/). This topological dual is always a [Banach space](https://www.edgepedia.org/banach-space), even when \( X \) itself is not complete[4](https://www.buttenschoen.ca/MATH725/duality/duality/).
In infinite dimensions the two notions separate: the algebraic dual strictly contains the topological dual, and the discontinuous functionals it adds are constructed using a Hamel basis via the axiom of choice[4](https://www.buttenschoen.ca/MATH725/duality/duality/). For any finite-dimensional [normed vector space](https://www.edgepedia.org/normed-vector-space), by contrast, the algebraic dual coincides with the topological dual[9](https://ncatlab.org/nlab/files/deBoor-LinearAlgebra.pdf).
## History
The word "duality" entered mathematics in 1826, when Gergonne introduced it for the point-line reciprocity of projective geometry[10](https://www.math.purdue.edu/~gottlieb/Bibliography/53.pdf). The linear-algebraic dual space is a later refinement. According to [Jean Dieudonné](https://www.edgepedia.org/jean-dieudonne), before 1930 nobody had a correct conception of duality between finite-dimensional vector spaces; even in van der Waerden's 1931 textbook, a vector space and its dual were still identified rather than kept distinct[11](https://math.stackexchange.com/questions/188371/who-was-the-first-to-use-dual-space).
## References
1. [Santiago Cañez, Notes on Dual Spaces, Northwestern University](https://sites.math.northwestern.edu/scg479/courses/notes/dual-spaces.pdf)
2. [David Wilkins, Section 2.3: Dual Spaces, MAU34802 Course Notes, Trinity College Dublin](https://www.maths.tcd.ie/~dwilkins/Courses/MAU34802/MAU34802_Hil2021_CourseNotes/MAU34802_Hil2021_Section2.pdf)
3. [Noam Elkies, Math 55a: Duality basics, Harvard University](https://people.math.harvard.edu/~elkies/M55a.02/dual.html)
4. [The Dual Space, Applied Functional Analysis (MATH 725), University of British Columbia Okanagan](https://www.buttenschoen.ca/MATH725/duality/duality/)
5. [Frank Narcowich, Math 641-600 Notes: Vector Spaces, Texas A&M University](https://people.tamu.edu/~f-narcowich/m641/m641_notes/vector_spaces.html)
6. [The Dual of a Vector Space, Math 5101 Lecture Notes, Ohio State University](https://people.math.osu.edu/gerlach.1/math5101/5101lecturenotes/DualOfAVectorSpace.pdf)
7. [Daniel Miranda, Espaços Duais, Álgebra Linear Avançada, UFABC](https://danielmiranda.prof.ufabc.edu.br/livros/ala/capitulo-05-espacos-duais.html)
8. [4.1 Dual Spaces and Bilinear Forms, Algebra General Exam Review, University of Virginia](https://math.virginia.edu/graduate/exams/algebra/alg-prep-4-1.html)
9. [Carl de Boor, The Dual of a Vector Space](https://ncatlab.org/nlab/files/deBoor-LinearAlgebra.pdf)
10. [James C. Becker and Daniel Henry Gottlieb, A History of Duality in Algebraic Topology, Purdue University](https://www.math.purdue.edu/~gottlieb/Bibliography/53.pdf)
11. [Who was the first to use dual space?, Mathematics Stack Exchange](https://math.stackexchange.com/questions/188371/who-was-the-first-to-use-dual-space)
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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Linear and multilinear algebra › Vector spaces and linear maps*
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