Dual Space · aiwiki.page

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)