The dual space of a vector space is the vector space of scalar-valued linear functionals on it. A functional assigns a scalar to each vector while preserving addition and scalar multiplication. In linear algebra, the dual contains all such functionals; in functional analysis, “dual space” usually means the continuous dual, containing only continuous linear functionals. These constructions distinguish vectors from the linear measurements performed on them. (stat.uchicago.edu)
Algebraic definition
Let (V) be a vector space over a field (\mathbb F). Its algebraic dual is [ V^*=\operatorname{Hom}_{\mathbb F}(V,\mathbb F). ] A member (\varphi), called a linear functional or covector, is a linear map satisfying [ \varphi(av+bw)=a\varphi(v)+b\varphi(w). ] Addition and scalar multiplication are defined pointwise: [ (\varphi+\psi)(v)=\varphi(v)+\psi(v),\qquad (a\varphi)(v)=a\varphi(v). ] The evaluation pairing ((\varphi,v)\mapsto\varphi(v)) is linear in each argument and requires no inner product. (stat.uchicago.edu)
For example, on (\mathbb F^n), every functional has the form [ \varphi(x)=a_1x_1+\cdots+a_nx_n. ] With vectors represented by columns, functionals can therefore be represented by rows. The expression resembles a dot product, but its definition is algebraic: over the complex field, no complex conjugation occurs in this pairing. (linear.axler.net)
Dual bases and coordinates
If (V) has a finite basis (e_1,\ldots,e_n), its corresponding dual basis (e^1,\ldots,e^n) is uniquely characterized by [ e^i(e_j)=\delta_{ij}, ] where (\delta_{ij}) equals one when (i=j) and zero otherwise. Thus (e^i) extracts the (i)-th coordinate. If (v=\sum_i x_i e_i), then [ e^i(v)=x_i,\qquad \varphi=\sum_i\varphi(e_i)e^i. ] Consequently, (\dim V^*=\dim V), so a finite-dimensional space and its dual are isomorphic. An identification obtained by sending (e_i) to (e^i), however, depends on the chosen basis. (stat.uchicago.edu)
If new basis vectors are related to the old ones by an invertible matrix (P), vector coordinates transform by (P^{-1}), whereas functional row coordinates transform by (P). These opposite transformations preserve the scalar (\varphi(v)). In infinite dimensions, the coordinate functionals associated with an algebraic basis generally do not span the whole algebraic dual: specifying a functional permits arbitrary values on all basis vectors, not merely finitely many nonzero values. (linear.axler.net)
Dual maps and annihilators
A linear map (T:V\to W) induces a map in the opposite direction, [ T^:W^\to V^,\qquad T^(\psi)=\psi\circ T. ] This is the dual map, also called the transpose map. It pulls a functional on (W) back to one on (V). If (T) has matrix (A), its dual map has the transpose matrix (A^{\mathsf T}) in the corresponding dual bases. Composition reverses: [ (S\circ T)^=T^\circ S^*. ] This algebraic transpose should not be confused with the conjugate-transpose matrix used for complex inner-product adjoints. (linear.axler.net)
For a subspace (U\subseteq V), its annihilator is [ U^\circ={\varphi\in V^*:\varphi(u)=0 \text{ for every }u\in U}. ] In finite dimensions, [ \dim U^\circ=\dim V-\dim U. ] Functionals on the quotient space (V/U) correspond naturally to members of (U^\circ), because precisely these functionals are constant on cosets of (U). (linear.axler.net)
The double dual
The dual of (V^) is the double dual (V^{}). There is a canonical evaluation map [ J:V\to V^{},\qquad J(v)(\varphi)=\varphi(v). ] Unlike a basis-dependent identification with (V^), this construction makes no coordinate choices. The map is injective because linear functionals separate distinct vectors. In finite dimensions, equality of dimensions makes (J) surjective as well, giving a canonical isomorphism (V\cong V^{**}). (stat.uchicago.edu)
For the continuous dual of a normed vector space, the Hahn–Banach theorem ensures that (J) is an isometric embedding. A Banach space is called reflexive when this canonical embedding is onto. Reflexivity is therefore stronger than merely having some isomorphism with the double dual. (ocw.mit.edu)
Continuous duals and representation
For a normed space (X) over the real or complex field, the continuous dual consists of bounded linear functionals. Its norm is [ |\varphi|{X^*} =\sup{|x|\le1}|\varphi(x)|. ] This dual is always a Banach space, even if (X) is incomplete. In finite-dimensional normed spaces, every linear functional is continuous; in infinite dimensions, the continuous dual can be a proper subspace of the algebraic dual. (ocw.mit.edu)
For a Hilbert space (H), the Riesz representation theorem states that every continuous linear functional has a unique representation [ \varphi(x)=\langle x,y\rangle,\qquad |\varphi|=|y|. ] Here the inner product is taken to be linear in its first argument. Over the complex field, (y\mapsto\varphi) is conjugate-linear; over the real field, it is linear. Thus this identification uses additional inner-product structure, rather than belonging to every vector space intrinsically. (ocw.mit.edu)
On a finite measure space, the continuous dual of an \(L^p\) space, for (1\le p<\infty), is represented by (L^q), where (1/p+1/q=1), through integration against a function. For (1<p<\infty), these spaces are reflexive. Duals also define the weak topology on (X) and the weak-star topology on (X^*): convergence is tested, respectively, by every continuous linear functional and by evaluation at every vector of (X). (math.mit.edu)