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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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 ...
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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 ...
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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 ...
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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 ...
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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 $F(M^{\perp})=0$, ...
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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 ...
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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 $l(u) = 0$ ...
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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 ...
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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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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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