Linear Functional · aiwiki.page

A linear functional is a linear map from a vector space to its underlying field of scalars. Unlike a general linear operator, which may return a vector in another space, a linear functional returns a single scalar. The terminology applies both to finite-dimensional spaces and to spaces whose elements are functions. Linear functionals form the dual space and provide a framework for studying coordinates, scalar measurements, and continuity. (pi.math.cornell.edu)

Definition and elementary properties

Let $V$ be a vector space over a field $\mathbb F$ . A map $f:V\to\mathbb F$ is a linear functional if

f(\alpha x+\beta y)=\alpha f(x)+\beta f(y)

for every $x,y\in V$ and $\alpha,\beta\in\mathbb F$ . This combines additivity and homogeneity. It implies $f(0)=0$ , $f(-x)=-f(x)$ , and preservation of every finite linear combination. In analysis, the scalars are usually real numbers or complex numbers. (pi.math.cornell.edu)

Linearity depends on the specified scalar field: a real-linear map on a complex vector space need not be complex-linear. In particular, complex conjugation is real-linear but not complex-linear. Also, an expression $x\mapsto f(x)+b$ , where $b\ne0$ , is an affine map, not a linear functional, because it does not send zero to zero. (math.mit.edu)

Coordinates and the algebraic dual

On $\mathbb F^n$ , every linear functional has the form

f(x_1,\ldots,x_n)=a_1x_1+\cdots+a_nx_n.

Relative to the standard coordinates, it is represented by the $1\times n$ matrix $(a_1,\ldots,a_n)$ , acting on a column vector. Its coefficients determine the functional uniquely. (cs.cornell.edu)

More generally, if $e_1,\ldots,e_n$ is a basis of $V$ , its dual basis consists of functionals $e^1,\ldots,e^n$ satisfying

e^i(e_j)=\delta_{ij}.

Thus, if $x=\sum_jx_je_j$ , then $e^i(x)=x_i$ , and

f=\sum_{i=1}^{n}f(e_i)e^i.

The algebraic dual $V^*=\operatorname{Hom}_{\mathbb F}(V,\mathbb F)$ , with pointwise addition and scalar multiplication, consequently has the same dimension as $V$ when $V$ is finite-dimensional. (math.mit.edu)

A choice of basis yields an isomorphism between $V$ and $V^*$ , but the vector-space structure alone does not specify one canonically. By contrast, the evaluation map into the double dual,

J:V\longrightarrow V^{**},\qquad J(x)(f)=f(x),

requires no basis and is an isomorphism in finite dimensions. (math.mit.edu)

Kernels and hyperplanes

The kernel of a functional is

\ker f=\{x\in V:f(x)=0\},

a linear subspace. If $f\ne0$ , then $f$ is surjective onto $\mathbb F$ , and its kernel has codimension one. In finite dimensions, this follows from the rank–nullity theorem. Such a kernel is a linear hyperplane. (homepages.ucl.ac.uk)

Choose $v$ with $f(v)=1$ . Every vector admits the decomposition

x=\bigl(x-f(x)v\bigr)+f(x)v,

where the first summand lies in $\ker f$ . For any scalar $c$ , the level set $f(x)=c$ is the translated hyperplane $cv+\ker f$ . Over the real numbers, these level sets describe parallel scalar-valued constraints. (homepages.ucl.ac.uk)

Function-space examples

A fixed-point evaluation is linear: on the space of continuous functions on $[a,b]$ ,

E_t(g)=g(t),\qquad t\in[a,b].

Likewise, the integration functional

I(g)=\int_a^b g(s)\,ds

is linear. Weighted integration, $g\mapsto\int_a^b g(s)w(s)\,ds$ , gives further examples whenever the indicated integrals exist. (pi.math.cornell.edu)

On the space of polynomials, extracting a specified coefficient is a linear functional. Evaluating a derivative at a fixed point, $p\mapsto p'(t)$ , is another example. These illustrate why “functional” does not imply nonlinearity or require an infinite-dimensional domain: the defining feature is scalar-valued linearity. (cs.cornell.edu)

Continuity and boundedness

In functional analysis, the domain often carries a norm. A functional on a normed vector space $X$ is continuous exactly when there exists $C\ge0$ such that

|f(x)|\le C\|x\|\qquad(x\in X).

Continuity at zero alone suffices. Here, “bounded” means bounded on the unit ball, not bounded on the entire space; any nonzero functional grows without bound along suitable scalar multiples. (ocw.mit.edu)

The functional’s operator norm is

\|f\|=\sup_{\|x\|\le1}|f(x)|.

The continuous dual consists only of continuous linear functionals, unlike the algebraic dual, which imposes no continuity requirement. The continuous dual is a Banach space under this norm even if $X$ is incomplete. Every functional on a finite-dimensional real or complex normed space is continuous; infinite-dimensional spaces can have discontinuous functionals. (ocw.mit.edu)

Representation, extension, and weak convergence

For a Hilbert space $H$ , the Riesz representation theorem states that every continuous linear functional has a unique representation

f(x)=\langle x,y\rangle,\qquad y\in H,

using the convention that the inner product is linear in its first argument. Moreover, $\|f\|=\|y\|$ . In complex spaces, the correspondence $y\mapsto f$ is conjugate-linear. (ocw.mit.edu)

The Hahn–Banach theorem ensures that a bounded linear functional on a subspace of a real or complex normed space extends to the whole space without increasing its norm. Consequently, continuous functionals separate distinct points and recover the norm through

\|x\|=\sup_{\|f\|\le1}|f(x)|.

Extensions need not be unique. (ocw.mit.edu)

Linear functionals also define the weak topology: the weakest topology making every member of the continuous dual continuous. A sequence $x_n$ converges weakly to $x$ precisely when $f(x_n)\to f(x)$ for every continuous linear functional $f$ . This tests convergence through scalar evaluations rather than directly through norm distances. (math.ucdavis.edu)