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Chapter 3: Problem 7

Let \(X\) be a compact Hausdorff space. A positive linear functional on \(C(X)\) is a (bounded) linear functional \(A\) with the additional property that \(A(f) \geq 0\) whenever \(f \geq 0\) on \(X\). Show that point evaluation at \(x_{0} \in X\) is a positive linear functional for each \(x_{0} \in X\). There is a representation theorem for the positive linear functionals on \(C(X)\) which says that for each positive linear functional \(\Lambda\) there is a unique positive, finite, regular Borel measure \(\mu\) on \(X\) with $$ \Lambda(f)=\int_{X} f d \mu . $$ (See Section A.5 in the Appendix for further discussion). If \(A\) is point evaluation at \(x_{0}\), what is the corresponding measure \(\mu\) ?

Short Answer

Expert verified
The corresponding measure \(\mu\) for point evaluation at \(x_0\) is the Dirac measure at \(x_0\).

Step by step solution

01

Define the Problem

We aim to establish that the point evaluation functional at a point \(x_0\) in a compact Hausdorff space \(X\) is a positive linear functional. This means we need to prove it is linear and positive.
02

Define Point Evaluation Functional

The point evaluation functional at \(x_0\), denoted \(A_{x_0}\), is defined by \(A_{x_0}(f) = f(x_0)\) for every \(f \in C(X)\).
03

Check Linearity of Point Evaluation

To prove linearity, consider two functions \(f, g \in C(X)\) and scalars \(a, b \in \mathbb{R}\). Calculate: \[ A_{x_0}(af + bg) = (af + bg)(x_0) = af(x_0) + bg(x_0) = aA_{x_0}(f) + bA_{x_0}(g).\] Hence, \(A_{x_0}\) is linear.
04

Establish Positivity of Point Evaluation

To confirm positivity, assume \(f \geq 0\) on \(X\). Then, \(f(x_0) \geq 0\). Therefore, \[ A_{x_0}(f) = f(x_0) \geq 0. \] Thus, \(A_{x_0}\) is a positive linear functional.
05

Connect Positive Linear Functional to Representation Theorem

By the given representation theorem, each positive linear functional corresponds to a unique positive, finite, regular Borel measure \(\mu\) such that \[ \Lambda(f) = \int_X f \, d\mu. \]
06

Determine the Corresponding Measure \(\mu\)

For point evaluation \(A_{x_0}\), the measure \(\mu\) is a Dirac measure centered at \(x_0\). Specifically, for a measurable set \(E \subset X\), \[ \mu(E) = \begin{cases} 1, & \text{if } x_0 \in E, \ 0, & \text{if } x_0 otin E. \end{cases} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Positive Linear Functional
A positive linear functional is an essential concept in functional analysis. It is a special kind of linear functional. A linear functional is a function that maps functions to real numbers while maintaining linearity. This means that it respects addition and scalar multiplication. In mathematical terms, if we have functions \(f\) and \(g\) in a linear space and scalars \(a\) and \(b\), a linear functional \(A\) satisfies:
  • \(A(af + bg) = aA(f) + bA(g)\)
A positive linear functional has the added requirement that it produces non-negative values for non-negative input functions. For a compact Hausdorff space \(X\), the space \(C(X)\) refers to continuous functions defined on \(X\). If \(A\) is a positive linear functional on \(C(X)\) and \(f \geq 0\), then \(A(f) \geq 0\). This positivity ensures that the image of non-negative functions under \(A\) remains non-negative. This property makes positive linear functionals very useful for establishing connections with measures, especially when applied to spaces like \(C(X)\).
Compact Hausdorff Space
A Compact Hausdorff Space is an interesting and widely applicable type of topological space. To understand it, we need to break down the terms 'compact' and 'Hausdorff'. A space is compact if every open cover has a finite subcover. In simple terms, if you have a collection of open sets that cover the space, you can find a finite number of these sets that still cover the entire space. This property is crucial in many mathematical contexts, as it can imply boundedness and finiteness in broader, more abstract senses. Hausdorff refers to a separation property of the space. A space is Hausdorff if for any two distinct points, there exist neighborhoods around each point that do not overlap. This is a strong requirement and is crucial for making precise distinctions between points. The Hausdorff property ensures that limits of sequences or nets are unique, which is fundamental in analysis. When combined, a compact Hausdorff space offers a powerful setting where many desirable properties of both compact and Hausdorff spaces are present. Such spaces often provide the context for defining continuous functions for which positive linear functionals are evaluated.
Dirac Measure
The Dirac Measure is a simple yet powerful concept in measure theory. It is quite straightforward, making it an ideal illustrative tool for many theoretical applications.The Dirac measure \( \delta_{x_0} \) is centered at a particular point \( x_0 \) in a space. It is defined as a measure that assigns a value of 1 to any measurable set containing \( x_0 \), and 0 to any set that doesn't. Mathematically, for a set \( E \), the Dirac measure is represented as:
  • \( \mu(E) = 1 \) if \( x_0 \in E \)
  • \( \mu(E) = 0 \) if \( x_0 otin E \)
This measure is used to represent the concept of the evaluation of functions at a point, linking to the idea of pointwise operations.In the context of the representation theorem for positive linear functionals, the Dirac measure provides the unique measure corresponding to the point evaluation functional. This makes it an invaluable concept when studying function spaces such as \(C(X)\) in functional analysis.
Point Evaluation Functional
The Point Evaluation Functional is a concept that ties directly into the assessment of functions at specific points. For a space of continuous functions, \(C(X)\), defined over a compact Hausdorff space \(X\), this functional plays a significant role.Point evaluation functional at a point \(x_0\), often denoted as \(A_{x_0}\), involves taking a function \(f\) and evaluating it at \(x_0\). This is expressed as \(A_{x_0}(f) = f(x_0)\). Its importance is highlighted in determining the properties of functions at specific points, serving various analytical tasks.The point evaluation functional is inherently linear. Given any two functions \(f\) and \(g\), and any scalars \(a\) and \(b\), the linearity property states:
  • \(A_{x_0}(af + bg) = aA_{x_0}(f) + bA_{x_0}(g)\)
Additionally, it is positive because if \(f \geq 0\), then \(f(x_0) \geq 0\), ensuring \(A_{x_0}(f) \geq 0\).These properties make point evaluation functionals easy to use within the framework of the Riesz Representation Theorem, connecting them to Dirac measures. This understanding is crucial when analyzing functions on compact Hausdorff spaces.

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Most popular questions from this chapter

Let \(c\) denote the linear subspace of \(\ell^{\infty}\) consisting of all sequences \(x=\left\\{x_{n}\right\\}_{1}^{\infty}\) for which \(\lim _{n \rightarrow \infty} x_{n}\) exists. (a) Let \(e=(1,1,1, \ldots) \in c\). Show that $$ c=\left\\{x+\alpha e: x \in c_{0} \text { and } \alpha \in \mathbb{C}\right\\} . $$ (b) Argue that the formula \(\varphi_{\infty}\left(\left\\{x_{n}\right\\}\right)=\lim _{n \rightarrow \infty} x_{n}\) defines a bounded linear functional on \(c\), where \(c\) is equipped with the supremum norm. (c) Show that \(c\) is a closed subspace of \(\ell^{\infty}\). (d) Given \(b=\left\\{b_{n}\right\\}\) in \(\ell^{1}\) and \(\gamma \in \mathbb{C}\), consider the linear functional defined on \(c\) by $$ \psi_{b, \gamma}\left(\left\\{x_{n}\right\\}\right)=\sum_{n=1}^{\infty} b_{n} x_{n}+\gamma \lim _{n \rightarrow \infty} x_{n} . $$ Show that the map \((b, \gamma) \rightarrow \psi_{b, \gamma}\) is an isometric isomorphism of \(\ell^{1} \times \mathbb{C}\), equipped with the norm \(\|(b, \gamma)\|=\|b\|_{1}+|\gamma|\), onto \(c^{*}\).

The point of this problem is to show that Theorem \(3.11\) may fail in a normed linear space that is not a Banach space. Let \(F\) be the set of "eventually zero" sequences, in the supremum norm; this means that a sequence \(\left\\{a_{n}\right\\} \in \ell^{\infty}\) belongs to \(F\) if there is an \(N\) with \(a_{n}=0\) for all \(n \geq N\). Define linear maps \(T_{n}: F \rightarrow \mathbb{C}\) by $$ T_{n}\left(\left\\{a_{k}\right\\}\right)=\sum_{k=1}^{n} a_{k} . $$ Show that each \(T_{n}\) is linear and bounded and for any fixed sequence \(x=\left\\{a_{k}\right\\}\) in \(F\), \(\sup \left\\{\left|T_{n}(x)\right|: n=1,2,3, \ldots\right\\}\) is finite. Is \(\sup \left\\{\left\|T_{n}\right\|: n=1,2,3 \ldots\right\\}<\infty\) ?

Suppose that \(\varphi: \mathbb{D} \rightarrow \mathbb{D}\) is an analytic function (where \(\mathbb{D}\) is the open unit disk) with the property that \(f \in L_{a}^{2}(\mathbb{D})\) implies \(f \circ \varphi \in L_{a}^{2}(\mathbb{D})\). Define \(C_{\varphi}: L_{a}^{2}(\mathbb{D}) \rightarrow\) \(L_{a}^{2}(\mathbb{D})\) by \(C_{\varphi}(f)=f \circ \varphi\). Show that the composition operator \(C_{\varphi}\) is a bounded linear operator on \(L_{a}^{2}(\mathbb{D})\).

Show that if \(X\) is a Banach space that is not reflexive, then \(X^{*}\) is also not reflexive. Hint: Find a nonzero bounded linear functional on \(X^{* *}\) which is 0 on \(\left\\{x^{* *}: x \in X\right\\}\). The converse statement is also true; see p. 132 in [8].

A sequence \(\left\\{h_{n}\right\\}\) of vectors in a Hilbert space \(\mathscr{H}\) is said to be a Bessel sequence if $$ \sum_{n=1}^{\infty}\left|\left\langle h, h_{n}\right\rangle\right|^{2}<\infty $$ for every \(h \in \mathscr{H}\). A sequence \(\left\\{g_{n}\right\\}\) is said to be a Riesz-Fischer sequence if given any \(\left\\{c_{n}\right\\} \in \ell^{2}\) there exists (at least one) vector \(g \in \mathscr{H}\) such that $$ \left\langle g, g_{n}\right\rangle=c_{n} \text { for all } n $$ Note that an orthonormal basis is both a Bessel sequence and a Riesz-Fischer sequence. (a) Show that if \(\left\\{h_{n}\right\\}\) is a Bessel sequence, then there exists \(M<\infty\) so that $$ \sum_{n=1}^{\infty}\left|\left\langle h, h_{n}\right\rangle\right|^{2} \leq M\|h\|^{2} $$ for all \(h \in \mathscr{H}\). Hint: Apply the closed graph theorem to the map \(S: \mathscr{H} \rightarrow \ell^{2}\) defined by \(S h=\left\\{\left\langle h, h_{n}\right\rangle\right\\}\). (b) Show that if \(\left\\{g_{n}\right\\}\) is a Riesz-Fischer sequence, there exists \(m>0\) such that given \(\left\\{c_{n}\right\\} \in \ell^{2}\), the equations in (3.5) hold for at least one solution \(g\) satisfying $$ m\|g\|^{2} \leq \sum_{n=1}^{\infty}\left|c_{n}\right|^{2} . $$ Hint: The closed graph theorem again, applied to the appropriate map $$ T: \ell^{2} \rightarrow \mathscr{H} / N $$ where \(N\) is the orthogonal complement of the closed linear span of the vectors \(g_{n}\).
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