# 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).

293 questions
Filter by
Sorted by
Tagged with
32 views

### 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 ...
1 vote
28 views

### 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 $F(M^{\perp})=0$, ...
1 vote
28 views

### 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 ...
56 views

### 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 ...
63 views

56 views

38 views

### 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 $l(u) = 0$ ...
41 views

### $BV_{\text{loc}}(U)$ 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 ...
136 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 ...
37 views

1 vote
40 views

### 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-...
115 views

### 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 $X$. $(Y, \mathcal Y)$ ...
27 views

72 views

32 views

### 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 ...
79 views

### 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 ...
60 views

1 vote
74 views

### 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 ...
76 views

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 $\... 0 votes 0 answers 39 views ### Riesz representation theorem$\mathbb{P}_2$What I have given is the Riesz representation theorem. I don't really understand how it works. Let$\mathbb{P}_2$be the vector space of polynomials with the highest degree of 2. Find the riesz ... 2 votes 1 answer 119 views ### How to describe the Gelfand transform of the Banach algebra of complex Borel measures on the real line? Let$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)... 73 views

### 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 ...
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\...
### 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)\...