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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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 ...
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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 $X$....
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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 ...
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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 + \...
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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 ...
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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 ...
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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 ...
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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\|...
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2 votes
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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 ...
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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 \ ...
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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(\...
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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}...
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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 ...
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Linear operators Hilbert space associated to bilinear operators

Let $H_1,H_2$ be two Hilbert spaces. Then it is often used that $${L}(H_1;{L}(H_1,H_2))\simeq {BL}(H_1\times H_1;H_2)$$ where $L$ denotes the space of linear operators and $BL$ the space of bilinear ...
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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 $\...
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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 ...
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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 ...
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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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Representing sums of linear functionals on locally compact Hausdorff space using Riesz representation theorem.

Suppose $\Lambda_1, \Lambda_2$ are positive linear functionals on $C_c(X)$, where $X$ a locally compact Hausdorff space $X$ then by Riesz Representation theorem for each one of them there's a sigma ...
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Rudin Real and Complex analysis - Step II theorem 2.14, Riesz

Trying to understand this very long theorem of which I think good understanding is very educational. I am going through all the steps specifically now there's a subtlety in the conclusion of step II ...
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1 answer
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John Conway's proof of Riesz representation theorem

I'm studying Functional Analysis from John Conway's "A Course in Functional Analysis" and I needed to go over some things from the first chapter and decided to reread the proof of Riesz ...
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Questions in the Proof of Riesz Representation Theorem for $C_0(X)$

I'm going through the proof of Riesz Representation Theorem for $C_0(X)$, namely Theorem $6.19$ in Rudin's Real and Complex Analysis. I have the following questions from the theorem's proof, which ...
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Prove a Borel measure coincides with a Riesz measure on Borel $\sigma$-algebra

I try to prove the Lemma used here :Borel measure and Riesz measure To prove: If a Borel measure $\mu$ coincides with a Riesz measure $\lambda$ on any open set in $\mathbb{R^n}$, then they coincides ...
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Stampacchia's theorem (proof in Brezis's book)

I was studying the proof of Stampacchia's theorem from Haim Brezis's book, attached the theorem with the proof: Theorem: Assume that $a(u,v)$ is a continuous coercive bilinear form on $H$. Let $K\...
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1 answer
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Problem $2.17$, Rudin's RCA (Dictionary Order Topology)

Problem $2.17$: Define the distance between the points $(x_1,y_1)$ and $(x_2,y_2)$ in the plane to be $$|y_1-y_2| \quad \text{if }x_1 = x_2, \quad\quad 1+|y_1 - y_2|\quad \text{if } x_1\ne x_2$$ Show ...
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Compactness of set of measures with respect to weak-*topology

I am reading a proof of the Stone-Weierstrass theorem by De Branges, but I am having trouble understanding the following part. Let $E$ be a locally compact Hausdorff space and $C_0(E, \mathbb{C})$ the ...
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Given a non-regular measure, working with a regular one, and going back.

I'm trying to show the following. Proposition. Let $T$ be the unit circle in the complex plane. Let $\mu$ be a complex Borel measure on $T$. Define the sequence $\hat\mu$ by $$ \hat\mu(n) := \int e^{-...
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2 votes
1 answer
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A monotonically increasing function $g$ as a measure.

I have three simple questions. I'm working with a problem in an old qualification exam, which asks me to express $\mu(E)$ explicitly, within the settings. Settings. Let $g:[0,1]\to \mathbb R$ be be a ...
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Every function $f \in L^2(\Omega)$ admits a weak derivative.

I have write a little argument that seems to show that every function $f \in L^2(\Omega)$ admits a weak derivative: Let $f \in L^2(\Omega)$. We define the functional on $H^1(\Omega) = W^{1, 2}(\Omega)$...
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-1 votes
2 answers
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Riesz representation theorem proof

UPDATE: There's an error in the question I was given. I would appreciate it if you could help me with the following problem I have in one of my Linear Algebra course questions: We have an inner ...
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1 vote
1 answer
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Counter-Example Riesz Representation theorem Linear Algebra

Let V be the vector space of polynomials over the field of complex numbers, with the inner product $\langle f,g \rangle = \int_{0}^{1} f(t) \overline{g(t)}dt$. Let $L$ be a linear functional defined ...
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Is the evaluation functionals on $C(K)$ correspond to Dirac delta measures?

Suppose $K$ is a compact and $T_2$ space and $a \in K$. Define the linear map $$E_a:(C(K), ||.||_\infty) \to \Bbb{C}\\f \mapsto f(a)$$ Then $||E_a||= 1$. So by Riesz-Representation Theorem for ...
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Relationship of transpose of linear mapping and Riesz isomorphism [closed]

How do I show the following: Be $Z$ a limited Euclidean vector space with the inner product space $\langle \cdot,\cdot \rangle$ and $\Phi : Z \to Z^*, z \mapsto \langle \cdot,z \rangle$ (Riesz ...
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Representer theorem - dot product of 2 summations

The Representer theorem on the reproducing kernel Hilbert space has the form \begin{align*} \langle K(\cdot, x_i), f\rangle_{\mathcal{H}} = f(x_i) \end{align*} where $K$ is a reproducing kernel with ...
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Exercise 8.T Bartle

Let $(X,\mathbf{X},\mu)$ be a measure space. Define the linear bounded operator $$G(f)=\int_X gfd\mu,\,\forall f\in L_1(\mathbf{X},\mu)$$ where $g\in L_\infty(\mathbf{X},\mu)$. Take $E_c=\{x\in X:|g(x)...
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Computing the measure induced by a Riemann-Stieltjes integral

Summary: (1). Is the Statement below true? (2). If so, then how to complete my proof sketched below? (3). Is there any way to compute the integral term in the Statement more explicity? I'm trying to ...
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Zorn's Lemma in the proof of Riesz representation theorem

I was reading the wikipedia article of the representation theorem of Riesz–Fréchet. see: https://en.wikipedia.org/wiki/Riesz_representation_theorem In the proof of the theorem, they say "Using ...
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Simpler proof of the Lax-Milgram Theorem if A is symmetric

In "Numerical Approximation of Partial Differential Equations" the authors Alfio Quarteroni and Alberto Valli state in remark 5.1.1, that the Riesz Representation Theorem suffices to prove ...
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Theorem 2.14 in Rudin's RCA: What is meant by "(a) forces (b)"?

Theorem 2.14 is the Riesz Representation Theorem(about functionals on $C_c(X)$): Theorem 2.14. Let $X$ be a locally compact Hausdorff space, and let $\Lambda$ be a linear functional on $C_c(X)$. Then ...
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1 answer
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Borel Measures on $\mathbb{R}$

My question is the following: do there exist Borel measures $\mu$ and $\nu$ such that, given a differentiable function $f$, we have $$ f'(0) = \int_\mathbb{R} f d\mu - \int_\mathbb{R} f d\nu? $$ I've ...
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1 vote
1 answer
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A theorem on hermitian forms

In Lang's Linear Algebra the following theorem was mentioned: I am not sure if I truly understand the result. The author here is referring to the Representation theorem, so I wrote the following ...
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Riesz' representative $v\in H^1$ of the functional $H^1\ni u\mapsto\langle u,f\rangle_{L^2}$ for $f\in L^1\setminus H^1$

Let $d\in\mathbb N$ and $\Omega\subseteq\mathbb R^d$. If $f\in L^2(\Omega)$, then $$\langle u,\varphi\rangle:=\langle u,f\rangle_{L^2(\Omega)}\;\;\;\text{for }u\in H^1(\Omega)$$ is clearly a bounded ...
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3 votes
2 answers
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Rudin RCA Problem 6.4

I'm trying to solve the following Problem 6.4 from Rudin's RCA: Suppose that $1\leq p\leq \infty$, and $q$ is the exponent conjugate to $p$. Suppose that $\mu$ is a $\sigma$-finite measure and $g$ is ...
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Using Lax Milgram lemma and energy estimates on the real line

I just want to check something. I want to use the energy estimates on the real line for an elliptic operator $L$ acting on $L^2(\mathbb{R})$. (The energy estimates are related to the Lax-Mi https://...
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2 votes
1 answer
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Riesz Representation Theorem and Continuous Map

The following is similar to a problem I found on some analysis quals: $X, Y$ are compact metric spaces and $F: X \to Y$ is surjective and continuous, if there is a finite measure $\nu$ on the Borel ...
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Counter Example About $L^{\infty}$ function

Let $(\Omega,\Sigma,\mu)$ be a measure space. Let $1<p<\infty$ and $q={p\over p-1}.$ Let $f\in L^p(\Omega,\Sigma,\mu).$ Then by the Riesz Representation Theorem, we know that, $$||f||_p=\sup_{||...
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2 votes
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If $V \hookrightarrow H$ are Hilbert spaces and $f \in H'$ then $f \in V'$?

Let $H=(H, (\cdot, \cdot)_H)$ and $V=(V, (\cdot, \cdot)_V)$ be Hilbert spaces such that $V \hookrightarrow H$ that is $V$ is continuously embedding in $H$. By Riesz representation theorem we can ...
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A Banach space isomorphic to his dual is an Hilbert space? [duplicate]

By the Riesz representation theorem, we know that the Hilbert space $\mathcal{H}$ is isomorphic to his dual. Is the converse true ? Does the fact that a Banach space $E$ is isomorphic to his dual ...
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