# Representation of a linear functional

We review results of Haviland and Riesz on the reperesentation of a linear functional.

Definition 1 Let ${X}$ be a subset of ${\mathbb{R}^n}$ and ${C(X)}$ be algebra of continuous functions on ${X}$. A positive linear functional on ${C(X)}$ is a linear functional ${L}$ with ${L(f) \ge 0}$ for all ${f \in C(X)}$ such that ${f(a) \ge 0, \forall a \in X}$.

We recall Haviland’s result in \cite{Marshall2} (also see \cite{Ha1, Ha2}), with ${\mathbb{R}[x_1, \dots, x_n]}$ denotes the ring of real multivariable polynomials:

Theorem 2 (Haviland) For a linear functional ${L: \mathbb{R}[x_1, \dots, x_n]}$ and closed set ${K}$ in ${\mathbb{R}^n}$, the following are equivalent:

1. ${L}$ comes from a Borel measure on ${K}$, i.e., ${\exists}$ a Borel measure ${\mu}$ on ${K}$ such that, ${\forall f \in \mathbb{R}[x_1, \dots, x_n], L(f) = \int f d\mu.}$
2. ${L(f) \ge 0}$ holds for all ${f \in \mathbb{R}[x_1, \dots, x_n]}$ such that ${f \ge 0}$ on ${K}$.

In Haviland’s theorem, a positive linear functional extended from ring of real multivariable polynomials to larger subalgebra and this theorem can be derived as a consequence of the following Riesz Representation Theorem (see \cite[p. 77]{KS}):

Theorem 3 (Riesz Representation Theorem) Let ${X}$ be a locally compact Hausdorff space and let ${L: C_c(X) \rightarrow \mathbb{R}}$ be a positive linear functional. Then there exists a unique Borel measure ${\mu}$ on ${X}$ such that

$\displaystyle L(f) = \int f d\mu, \forall f \in C_c(X).$

${C_c(X)}$ is the algebra of continuous functions with compact support.

# Riesz Representation Theorem – 1

This paragraph we follow the book of Rudin [1]. On linear functionals, there is special relationship between integration and linear functionals. In ${L^1(\mu)}$, a vector space, for any positive measure ${\mu}$, the mapping

$\displaystyle f \mapsto \int_X f d\mu.$

example, let ${C([0,1])}$ be the set of all continuous functions on the unit interval ${I = [0,1]}$. Then

$\displaystyle \Lambda f = \int_0^1 f(x)dx \quad \ (f \in C([0,1])),$

has two properties:

• ${\Lambda(f + g) = \int_0^1[f(x) + g(x)]dx = \int_0^1f(x)dx + \int_0^1g(x)dx = \Lambda(f) + \Lambda(g)}$.
• ${\Lambda(c.f) =\int_0^1cf(x)dx = c\int_0^1f(x)dx= c.\Lambda(f)}$.

so it is a linear functional on ${C([0,1])}$. Moreover, this is a positive linear functional: if ${f \ge 0}$ then ${\Lambda(f) \ge 0}$.

Consider a segment ${(a,b) \subset I}$ and the class of all ${f \in C(I)}$ with ${0 \le f(x) \le 1, \forall x \in I}$ and ${f(x) = 0, \forall x \notin (a,b)}$ or support of ${f}$ is ${(a,b) \subset I}$. So we get ${\Lambda(f) = \int_a^b f(x)dx \le \int_a^bdx = b-a}$. It is important that the length of ${(a,b)}$ related to the values of the functional ${\Lambda}$.

There is an important theorem of F. Riesz, this illustrates to above event

Theorem 1 (F. Riesz) Let $X$ be a closed subset of $\mathbb{R}$. Then every positive linear functional ${\Lambda}$ on ${C(X)}$ there corresponds a finite positive Borel measure ${\mu}$ on ${X}$ such that

$\displaystyle \Lambda(f) = \int_X fd\mu \quad \ (f \in C(X)).$

References

[1] W. Rudin, Real and Complex analysis, Mc.Graw-Hill, 1970.