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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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 ...
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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
63 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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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
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1 answer
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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,...
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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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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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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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$BV_{\text{loc}}(U)$ 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
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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 ...
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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 $H = ...
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The bijection in Riesz–Markov–Kakutani theorem is a homeomorphism w.r.t. both norm and weak$^*$ 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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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-...
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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})...
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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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Generalization of Riesz Representation theorem for $L^p$ spaces

Riesz-Representation Theorem for $L^p$ spaces says the following Let $p \in [1,\infty]$, let $(X,\mu)$ be a measure space, let $T \in (L^{p}(X,\mu) )$ meaning that $T \; : \; L^p(X,\mu) \to \mathbb{R}...
Paul's user avatar
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Measures on manifolds and zero measure sets

On a differentiable manifold $M$ there is a standard notion of zero measure sets using charts ( a set $A\subseteq M$ has zero measure if for every chart $(U,\varphi)$ $\varphi(A\cap U)$ has zero ...
Sart00's user avatar
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Riesz Representation Theorem for continuous linear functionals $T: L_1(\mu , \mathbb{R}^N)\rightarrow \mathbb{R}$

I am working on a problem where I am dealing with a continuous linear functional $T: L_1(\mu , \mathbb{R}^N)\rightarrow \mathbb{R}$ (i.e., from integrable random vectors to the reals), for which I ...
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Confusion about Functional Derivative from Wikipedia

In the Wikipedia page and from its references, it states that given a functional $F:B \to \mathbb{R}$ (where B in my case is a Banach Space), its functional derivative is defined as $$ \frac{d}{d\...
Matthew Ferrell's user avatar
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$3$ versions of Riesz–Markov–Kakutani theorem

I'm reading RKM theorem from this lecture note by professor Tomasz Kochanek. I have no question here. This thread is to summarize $3$ versions of the theorem (in an increasing order of generality). I ...
Analyst's user avatar
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What does it mean for a signed measure to be "regular" in Riesz-Markov-Kakutani theorem?

I'm reading RKM theorem from this lecture note by professor Tomasz Kochanek. Theorem 3.23 (Riesz-Markov-Kakutani for $\left.C_{0}(\boldsymbol{X})^{*}\right)$. Let $X$ be a locally compact Hausdorff ...
Analyst's user avatar
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1 vote
1 answer
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Extreme points of the closed ball of regular measures

Let $X$ be a topological vector space and $E$ a subset of $X$. A point $x\in E$ is called a extreme point of $E$ if there is no proper line segment contained in $E$ which contains $x$. Let $K$ be a ...
Aligomez's user avatar
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Riesz representation for linear functionals on space of continuously differentiable functions

It is known (Riesz Theorem) that every linear functional $f$ on $X=C[a,b]$ can be represented as $\int_a^bx(t)dv(t)$ for all $x\in X$, where $v$ has bounded variation. Is it true that every linear ...
Random Number's user avatar
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Finding vector associated with a form of $L^2(\mathbb{R})$.

I need to solve an exercise that asks for the vector associated to a certain form of the Hilbert space $L^2(\mathbb{R})$. I am pretty sure this "associated vector" is related to Riesz's ...
coffee_pls's user avatar
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2 answers
785 views

Riesz-Markov theorem: any positive linear functional is continuous?

Below is the Riesz–Markov theorem that I take from here. Riesz–Markov theorem: Let $X$ be a locally compact Hausdorff space and $C_0(X)$ the space of continuous compactly supported functionals on $X$....
Analyst's user avatar
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3 votes
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understanding why Wasserstein is weak

I am reading Wasserstein GAN paper and in Appendix A, it says Let $\mathcal{X} \subseteq \mathbb{R}^d$ be a compact set (such as $[0, 1]^d$ the space of images). We define Prob($\mathcal{X}$) to be ...
MoneyBall's user avatar
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4 votes
1 answer
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Showing the polarization of (complex) quadratic form is sesquilinear. [duplicate]

Let $q: V\times V \to \mathbb{C}$ on a complex vector space $V$ be a quadratic form. Define $\tilde q$ by the polarization identity: $$ \begin{equation} \tilde q(\phi,\psi) = \frac{1}{4} [q(\phi + \...
IGY's user avatar
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measure from a linear functional (Riesz)

I'm reading this wikibook which states that, from a linear functional $\Lambda$, one can build a measure. Setting: $X$ is a locally compact space. Here are the steps. Let $\Lambda: C^0(X,R)\rightarrow ...
Laurent Claessens's user avatar
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Rudin's RCA: Redundant argument to show regularity?

Although I understand the proof of the theorem, but I wonder about why proof of $(a)$ needs such a complex progress. The key idea is that if we can show $\mu(V - E) < \epsilon$ for any $E$, then by ...
Jiya's user avatar
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Riesz's representation Theorem ($f(x)=\langle w,x \rangle$)

In Kreyszig's book we have the following version of the Riesz's Representation Theorem: Theorem 3.8-1 Every bounded linear functional $f$ on a Hilbert space $H$ can be represented in terms of the ...
Math23's user avatar
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How are we supposed to show existence of a delta function on a linear subspace of $C[0,1]$ without reference to the Riesz representation?

Royden leaves the following as an exercise: Let $X$ be a linear subspace of $C[0,1]$ that is closed as a subset of $L^2[0,1]$. $X$ is closed, and there is a constant $M$ such that $\|f\|_\infty\le M\|...
FShrike's user avatar
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2 votes
1 answer
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"Coordinatized" proof of the Riesz representation theorem in $l^p, 1<p<\infty$

The following is motivated by Problem 2 in Paul Halmos' "Hilbert Space Problem Book" (my edition is from 1967 by American Book - Van Nostrand - Reinhold, thanks to Ralf Pradella for lending ...
Matthias Hübner's user avatar
1 vote
2 answers
92 views

Zero integration for any function in $C_c(X)$ would imply that the measure is zero?

Let $X$ be any locally compact Hausdorff space, and $\mu$ be a positive finite regular (both inner and outer) measure on $X$ such that $\operatorname{supp}(\mu)$ is compact. We have that $$\int_X f \ ...
Carl Butcher's user avatar
3 votes
1 answer
294 views

Dual of the Sobolev space

It is well known that for a given bounded domain $\Omega$, the Sobolev space $W^{1,2}(\Omega)$ is a Hilbert space, which is the space given by $$ W^{1,2}(\Omega)=\{u\in L^2(\Omega):\nabla u\in L^2(\...
Mathguide's user avatar
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Generating a positive measure $\nu$ such that $\displaystyle\sum_{n=1}^{\infty}w(n)f(n)=\int_{\mathbb N}fd\nu$

Let $w:\mathbb N\to [0,\infty)$ continuous. For each $f:\mathbb N\to\mathbb C$ such that $\displaystyle\sum_{n=1}^{\infty}w(n)f(n)$ is absolutely convergent we define $\Lambda f=\displaystyle\sum_{n=1}...
Will Smith's user avatar
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1 answer
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Proof of set being a sigma-algebra

Suppose $K$ is a compact metric space and let $F$ be a continuous linear functional on $C(K)$ (here $C(K)$ denotes the set of continuous functions on $K$). One version of the Riesz representation ...
Daniel Cederberg's user avatar
2 votes
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Representation theorem on dual of bounded continuous functions

I'm working through the proof of the above representation. Let $C_{b}(X)$ be the space of bounded continuous functions on a normal Hausdorf space. I've proven that for any positive linear functional $\...
Mark's user avatar
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Riesz representation theorem $\mathbb{P}_2$

What I have given is the Riesz representation theorem. I don't really understand how it works. Let $\mathbb{P}_2$ be the vector space of polynomials with the highest degree of 2. Find the riesz ...
John.W's user avatar
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How to describe the Gelfand transform of the Banach algebra of complex Borel measures on the real line?

Let $M$ be the Banach algebra of all complex Borel measures on $\mathbb{R}$. To be clear, Norm: $\| \mu \| = |\mu|(\mathbb{R})$, where $|\mu|(E)$ is the total variance. Product: $(\mu \ast \lambda)(E)...
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Norm of function in $C_0(\Omega)$

Let $(\Omega,\mu)$ be a measure space. For $f\in L^p(\Omega)$, I know that $$\|f\|_p=\sup\left\{\int_\Omega fg\, d\mu:g\in L^q(\Omega), \|g\|_q\leq 1\right\}.$$ I'm wondering whether something similar ...
Guest's user avatar
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application of Riesz representation theorem into dense subset

Let $\mathcal{H}$ be a Hilbert space with an inner product $\langle\cdot,\cdot\rangle$ and $V\subset\mathcal{H}$ be a dense subspace. We already know that $$\mathcal{H}^*=\{\langle v,\cdot\rangle|v\in\...
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Continuous linear operator on $L^2(\Omega)$

Let $\Omega\subset\mathbb{R}^2$ be an open and bounded domain and consider a positive continuous linear operator $B:L^2(\Omega)\to L^2(\Omega)$ with the property that if $f\in L^{\infty}(\Omega)\...
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