Questions tagged [riesz-representation-theorem]

Any question about any version of the Riesz-representation Theorem for linear forms on Hilbert spaces Bilinear forms on Hilbert spaces are also concerned including linear forms on $L^p$ spaces and Riesz- Markov theorem as well which extend to linear forms on Wiener spaces (spaces of continuous functions).

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The compactness of Radon probability measure

Let X be a compact subspace of $\mathbb{R}^n$. Denote by $\mathcal{R}$ the space of Radon measures on X, and $\mathcal{P}$ the space of Radon probability measures on X. In my book, it says that $\...
Lilileaf's user avatar
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Question in the proof of the Riesz Representation theorem of non-negative functionals in geometric measure theory written by Leon Simon

The problems are from the proof of Theorem 1.5.12 in Leon Simon's book: Geometric Measure Theory Suppose $X$ is a locally compact Hausdorff space, $\mathcal{K}^{+}$ is the set of all non-negative ...
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A characterization for the Riesz representation theorem vector

Let $H$ a Hilbert space and $\varphi \in H'$ a continuous linear functional. By Riesz Representation Theorem we know there exists $u_\varphi \in H$ such that $$ \varphi(v) = \langle v, u_\varphi \...
Lucas Linhares's user avatar
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What is the dual space of $C^1_0(\Omega)$ for open $\Omega \subset \mathbb{R}^n$?

What is the dual space of $C^1_0(\Omega)$ for open $\Omega \subset \mathbb{R}^n$, where $\Omega$ is an open, possibly unbounded set, and $C^1_0(\Omega)$ is equipped with the supremum norm $||f||:=\...
1Rock's user avatar
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Question in the proof of the Riesz Representation theorem (locally compact Hausdorff case)

I'm reading the proof of Riesz Representation theorem (1.5.14) in Leon Simon's book: Geometric Measure Theory Previously I've shown that $|L(fe_{j})|\leq \int_{X}|f|d\mu=\|f\|_{L^{1}(\mu)}$ for all $...
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Application of Riesz Representation in Boundary Value Problems [closed]

Q) I need to solve the following question using riesz representation theorem. $$ u''(x) = 1+x \\ u(0) = u(1) = 0 $$ I am aware of riesz representation theorem, I searched through the theory online but ...
Atom Bomb's user avatar
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Riesz representation theorem with Stieltjes integral?

I am a bit new to functional analysis and I stumbled upon this problem that confuses me. Consider the space $X$ of bounded, non-decreasing, right-continuous functionals $F$ on $[0,1]$. Specifically, ...
Ruben van Beesten's user avatar
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1 answer
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The proof of the Riesz representation thoerem for non-negative functionals

I was reading Introduction to Geometric Metric Theory by Leon Simon. Here's the full statement of the Riesz representation theorem for non-negative functionals. Suppose $X$ is a locally compact ...
OneLamp's user avatar
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The dual $(L^\infty)^{*}$ is not $L^1$ by constructing example

The problem statement is the same as this post: $L^{\infty *}$ is not isomorphic to $L^1$ . Let $L^\infty = L^\infty(m)$, where $m$ is Lebesgue measure on $I=[0,1]$ . Show that there is a bounded ...
Nazono Sumiko's user avatar
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What guarantees that the adjoint of a suitable integral operator, e.g. a Hilbert-Schmidt operator, is again an integral operator with a kernel?

This is likely a silly question, but I was wondering if $T$ is some nice integral transform, e.g. a Hilbert-Schmidt integral operator, with an, say, $L^2(\mathbb{R}^n)$ kernel, what then guarantees ...
Cartesian Bear's user avatar
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Adjoint operator and random variable

Let $(\Omega,\mathcal{A},\mathbb{P})$ be a probability space and $\mathbb{D}$ a dense subset of $L^2(\Omega,\mathcal{A},\mathbb{P})$. I consider a linear map $D$ from $\mathbb{D}\subset L^2(\Omega)$ ...
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Proof of Riesz Representation Theorem

I was trying to understand the proof of RRT from my lecture notes, but I have trouble in understanding the following step. $$T : L^{q} \rightarrow (L^{p})'$$ For $1 \leq p < \infty$. I understand ...
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Constructing a Radon measure from the integral of a function w.r.t another Radon measure

The problem is to show Suppose that $\mu$ is a Radon measure on $X$, If $\phi \in L^1(\mu)$ and $\phi \geq 0$,then prove that $\nu(E)=\int_E \phi d\mu$ is a Radon measure. My attempt: We first show ...
Nazono Sumiko's user avatar
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Counter example for Riesz representation theorem when $p=\infty$.

The Riesz representation theorem holds for $1\leq p\leq\infty$. I wonder whether there is a counterexample. Here is my attempt: Consider $C_{\mathbb{R}}([-1,1])\subset L^{\infty}([-1,1])$. Define a ...
Hongyi ZHOU's user avatar
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Justifying steps in Riesz Representation theorem (local compact hausdorff space case)

I am reading the proof of Riesz Representation theorem(1.5.14) on Leon Simon's book:Geometric Measure Theory And I got stuck at the following higlighted part. I completely understand the note he made ...
mikeqwertyuiop's user avatar
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2 answers
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A doubt in solving a projection mapping and orthogonality question

Let $l^2= \{ (x_1,x_2,x_3,\dots):x_n \in \mathbb{R} \text{ for all } n \in \mathbb{N} \text{ and } \sum_{n=1}^{\infty} x_n^2 < \infty \}.$ For a sequence $(x_1,x_2,x_3,\dots) \in l^2,$ define $$\...
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Riesz vector of extended linear functionals

Exercise 9.23 of Advanced Linear Algebra by Steven Roman: Let $f$ be a nonzero linear functional on a subspace $S$ of a finite-dimensional inner product space $V$ and let $K=\text{ker}(f)$. Let $f(v)= ...
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Is Lax-Milgram theorem too trivial to be a theorem?

Question: Can Lax-Milgram theorem be restated as: If an inner product $[\cdot,\cdot]$ induces a norm equivalent* to the norm of the Hilbert space $H=(X,\langle\cdot,\cdot\rangle)$, then $H'=(X,[\cdot,\...
Szeto's user avatar
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Compute the total variation of a linear bounded functional

Let $\mu$ be a Radon measure in $\mathbb R^n$, if $L:C_C^{0}(\mathbb R^n , \mathbb R^m) \to \mathbb R$ be a linear functional, the total variation of $L$ is defined by $$|L|(A)= \sup \{ L(\varphi): \...
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Proof of the Riesz-Markov-Kakutani Theorem (Rudin RCA 6.19)

I'm currently studying the proof of Theorem 6.19 from Real and Complex Analysis by Walter Rudin, which concerns the Riesz-Markov-Kakutani Theorem. I'd like to verify my understanding as well as some ...
bayes2021's user avatar
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Uniqueness of the measure in the Riesz-Markov-Kakutani theorem (Rudin RCA 6.19)

I'm currently studying the proof of Theorem 6.19 from Real and Complex Analysis by Walter Rudin, which concerns the Riesz-Markov-Kakutani Theorem. My question is about understanding the proof of the ...
bayes2021's user avatar
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A proof of the Riesz-Markov for compact subsets of the complex numbers.

I had the following idea of how to give a simple proof of the Riesz-Markov-Kakutani representation theorem for subsets of the complex numbers and I wanted to know whether someone sees some principle ...
DerHutmacher's user avatar
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Isometry in the Riesz representation theorem

In a proof of the Riesz representation theorem, I saw the following argument concerning the isometry ($H$ is a Hilbert space and $H^{*}$ its topological dual) : Take $\varphi\in H^{*}$. There exists $...
coboy's user avatar
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4 votes
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156 views

Riesz-Markov theorem and positive linear functionals on real-valued continuous functions

Riesz-Markov theorem: Let $X$ be a locally compact Hausdorff space. For any continuous linear functional $\Psi$ on $C_0(X)$, there is a unique regular countably additive complex Borel measure $\mu$ on ...
ayphyros's user avatar
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Hypothesis for establishing that the measure given by the Riesz representation theorem is a probability measure

I am trying to study, through the Riesz representation theorem applied to a space of compactly supported continuous functions, what hypotheses must be met to establish that the measure given by the ...
ayphyros's user avatar
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Classical balayage

Suppose $G \subset \mathbb{C}$ is a bounded domain, such that $\partial G$ has positive capacity, and let $v$ be a unit mass measure compactly supported in $G$. Then the balayage problem is to find a ...
porridgemathematics's user avatar
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$(\ell^{1})^{*}$ isometrically isomorphic to $\ell^{\infty}$ as a corollary of Riesz representation Theorem.

I'm following "A Course in Functional Analysis, Conway" and as a corollary of the Riesz theorem (Example 5.9) he states what I have written in the title. If I consider $\mathbb{N}$ with the ...
Davide Modesto's user avatar
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V a IPS of finite dimension, $\phi : V \to F$ is a linear functional. Let $B = \{v_1...v_n\}$ an orthonormal basis for V.

Prove Riesz unique representation theorom: If $v \in V$ is a vector in $V \implies \exists ! u \in V$ that stasfies $\phi (v) = \;<v,u>$ $\mathbf {Hint}$: express $u$ as a linear combination of ...
MathStudent101's user avatar
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Is there a non-Riesz way to prove the existence of uniqueness of this norm-preserving linear extension on Hilbert space?

$M$ is a linear subspace of Hilbert space $H$, $f$ is a bounded linear functional on $M$. Proof that there exists a norm-preserving linear extension of $f$ to $H$ call $F$ such that $F(M^{\perp})=0$, ...
anyon's user avatar
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Is the dual of continuous functions with compact support identified with Radon or Baire measures?

I have come across two different versions of the Riesz-Markov theorem, one identifies the dual with Radon measures and the other with Baire measures. From Wikipedia: Let $X$ be a locally compact ...
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Doubt about the demonstration of the uniqueness of the Fourier transform of a measure

For a project at my University, I have to prove that in the space of real numbers, if the Fourier transform of a measure is zero, then the measure is zero. I found a post with a solution to this ...
Rodrigo Pari's user avatar
2 votes
1 answer
98 views

Riesz representation theorem for variational equation

In a paper on solving PDE's I recently encountered following setting. Let $X$ be Hilbert space and $a:X\times X \to \mathbb{R}$ be a bounded symmetric positive definite bilinearform and $L:X \to \...
mz _'s user avatar
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Is there any relationship between Gleason's theorem and Riesz representation theorem?

Gleason's Theorem Section 2.3.3 states that if a mapping $p$ satisfies several conditions, $p$ can be written as $p(E)=\text{tr}(\rho E)$ which is the inner product with a given density matrix. The ...
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3 votes
1 answer
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Sigma-algebra used in the theorem of Lebesgue-Radon-Nikodym of Rudin's Real and Complex Analysis

The theorem of Lebegue-Radon-Nikodym in page 121 of Rudin's Real and Complex Analysis reads as: Let $\mu$ be a positive $\sigma$-finite measure on a $\sigma$-algebra $\mathfrak{M}$ on a set $X$, and ...
Epsilon Away's user avatar
1 vote
1 answer
93 views

Find element in Riesz Representation Theorem in a Sobolev Space

I am trying to solve the following question: Let $H = H^1([0,1])$ and let $Tu = u(1)$. Let $\langle \cdot,\cdot \rangle_H$ denote the standard inner product in $H$. By the Riesz representation theorem,...
user82261's user avatar
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Duals In The Sense Of Riesz Representation Theorem

Can somebody help me understand this quote from Wikipedia: In mathematics, a dual wavelet is the dual to a wavelet. In general, the wavelet series generated by a square-integrable function will have ...
fweth's user avatar
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2 votes
1 answer
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Riesz representation theorem for vector-valued maps

Let $\phi \colon H \to \mathbb{R}^d$ be a linear map defined on a real Hilbert space $H$. Let $e_i$ denote the standard basis, and $\phi_i = e_i^\ast \circ \phi$ denotes the projection of $\phi$ onto ...
Drew Brady's user avatar
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Uniqueness of Riesz representation theory

I am trying to prove the uniqueness of the Riesz representation theory for the real analysis context. The question statement is Given $\alpha\in BV[a,b]$, show that there is a unique $\beta\in BV[a, ...
al2000's user avatar
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Why does the kernel and a one dimensional subspace span the Hilbert space when proving the Riesz lemma?

In Reed & Simon's text on functional analysis,, they state the Riesz lemma as For each $T \in H^*$, there is a unique $y_T \in H$ such that $T(x) = (y_T, x)$ for all $x \in H$. In addition $\|Y_T\...
CBBAM's user avatar
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Applying Radon-Nikodym and Riesz Representation Theorem to Prove a Proposition

I have a paper that I am working on. I wanted to know can a function $\nu$ which is absolutely continuous with respect to measure $\mu$ be represented as an integral where the function is having a ...
P Initiate's user avatar
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Riesz representation theorem for the dual of L^\infty and C(X)

Suppose $\{f_n\}$ are real valued nonnegative continuous functions on $\bar\Omega$, where $\Omega$ is a bounded open subset of $\mathbb{R}^N$. Moreover, say $\int_\Omega f_n\le 1$ for all n. Then $\{...
user100356's user avatar
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1 answer
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Proof of Riesz's representation theorem in Stein's Real Analysis

I'm having some trouble regarding a step in the proof as it appears in Stein's Real Analysis. How is that the construction of $u$ guarantees that $u \in S$?. I obviously tried seeing that $l(u) = 0$ ...
nicoyanovsky's user avatar
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1 answer
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$BV_{\text{loc}}(U)$ functions

I'm reading the chapter 5 of the book "Measure Theory and Fine Properties" and there's one thing I don't understand. Theorem 1 (page 167) goes as follows: The proof uses the Riesz ...
Raúl Filigrana Villalba's user avatar
3 votes
1 answer
172 views

Riesz–Markov–Kakutani representation for symmetric matrix-valued measures

Let $\Theta$ be a closed $d$-manifold (and hence a metric space with Borel-$\sigma$-algebra $\Sigma \subset 2^\Theta$), $S^d$ the set of real symmetric $d \times d$ matrices, which we can identify ...
ViktorStein's user avatar
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Use of Riesz Representation theorem here

I'm working through Conways Functional analysis book right now, and I got a bit stumped, I just want to make sure I'm approaching the problem correctly. The problem statement goes as follows: Let $H = ...
Joey's user avatar
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The bijection in Riesz–Markov–Kakutani theorem is a homeomorphism w.r.t. both norm and weak$^*$ topologies

Let $X$ be a topological space, $\mathcal M(X)$ the space of regular complex Borel measures on $X$. $E :=C_0 (X)$ the space of $\mathbb C$-valued continuous functions on $X$ vanishing at infinity. $E^...
Analyst's user avatar
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3 votes
1 answer
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Finite signed measures: reconcile different types of convergence

Let $X$ be a metric space, $\mathcal M(X)$ the space of all Borel signed measures on $X$, $\mathcal C_b(X)$ be the space of real-valued continuous functions, $\mathcal C_0(X)$ the space of real-...
Analyst's user avatar
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Disintegration theorem: how is $\mu_y$ a probability measure for $\nu$-a.e. $y\in Y$?

Theorem 4 of this blog entry of Terrence Tao states that: Let $X$ be a compact metric space, $\mathcal X$ its Borel $\sigma$-algebra, and $\mu$ a Borel probability measure on $X$. $(Y, \mathcal Y)$ ...
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Proof verification: proving particular functional in $(C_0(\mathbb{R},\mathbb{C}))^*$ must be $0$

Please correct the following proof. I am trying to show that a linear functional $T$ must be identically $0$ if its dual representation satisfies a particular property. Let $C_0(\mathbb{R},\mathbb{C})...
stowo's user avatar
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1 answer
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Proving that Two functionals are equal

The following is an exercise from the third edition of Functional Analysis by B V Limaye (p. 438): Let $f$ and $g$ be two continuous linear functionals on a Hilbert space $H$. $\|g\|=\|f\|$ and $g(x)...
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