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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17 views

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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Restricting a functional to a subset of a Hilbert Space

A functional $φ∈H^∗$ corresponds to some vector $x∈H$ via the Riesz Representation Theorem; if $M$ is a closed linear subspace of H, φ can be restricted to act on it; i.e. $\tilde φ∈M^∗$ with $\tilde ...
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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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17 views

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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71 views

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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29 views

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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Integral representation of functionals in calculus of variations

What is the reason why in calculus of variations it is usual to restrict to study integral functionals of the following type: $$F(u) := \int_{\Omega} f(x,u(x),\nabla u(x) )dx$$ with $u$ in some space ...
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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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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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70 views

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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51 views

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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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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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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Show that a function is in $L^2(\mathbb{R_{+}^3})$

Let u $\in C^2([0, \infty] \times\mathbb{R^3})$ be the solution to the heat equation $$ u_{tt}-\nabla^2 =0 \\ u(0,x)=0\\ u_t(0,x)=h(x) \in C_0(\mathbb{R^3}) $$ Suppose there exists a constant $C$ ...
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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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If $E$ in $x- axis$, then $\mu(E)=\infty$.

Here, in this metric I intuitively think x-axis is not involved in metric space definition.Hence , no involvement of points on x-axis .We are not bother by points on x-axis .I am not able to prove $\...
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What does it mean to say that a measure belongs to $H^{-1}(\mathbb{R^2})$?

One of the lemmas begins with the following sentences. Let us consider a bounded measure $\mu$ so that the measure $(1+\vert x \vert)\mu$ is also bounded. If moreover $\mu$ belongs to $H^{-1}(\mathbb{...
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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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42 views

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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54 views

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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94 views

What is the predual space of $H_0^1(\Omega)$?

I am trying to understand the predual space $X$ of $H_0^1(\Omega)$. My idea was to identify the predual space by the canonical embedding $i:X\to (H_0^1(\Omega))^{*}$. I know that a Hilbert space is ...
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Riesz Representation Theorem, true for pre-Hilbert spaces and any functional?

I have some doubts about the Riesz theorem. Firstly can you check my proof? Fa = for all Fa (H,<,>) a pre Hilbert space Fa x in H different from the zero vector Fa F a functional on H Fa T: (H-&...
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About a point in the the answer to “Riesz representation and vector-valued functions”.

In the Q&A Riesz representation and vector-valued functions, @anon gave an answer. why $m_{\phi}$ is well-defined? Could some one give some reference or explain more about it?
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An alternative proof for Riesz Representation Theorem for $C([0,1])$

During my class in real analysis, my teacher mentioned an alternative method to prove Riesz's theorem for $C([0,1])$. He just mentioned the method, but did not prove it. (A usual way of doing this is ...
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Proving that a Hilbert space $\mathcal{H}$ is isometrically isomorphism to $l_2(I)$.

Let $\mathcal{H}$ an arbitrary Hilbert space no necessary Separable. Let $\{u_i:i\in I\}$ a orthonormal basis of $\mathcal{H}$ where $I$ is a uncountable set. Let $\mathbb{K}$ a field and $l_2(I)$ the ...
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Existence and uniqueness of the adjoint of a linear map.

I have this proposition from Karlheinz Spindler's Abstract algebra with applications vol. 1, page 303. It is proved there, but I tried it by myself based on that prove. I would like to know if my ...
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Confusion about uniqueness of $\sigma$-algebra in Riesz-representation theorem

In Rudin's "real analysis and complex analysis" the Riesz representation theorem is proved and used to define the lebesgue measure on $\mathbb{R}^n$. I have a question about this. First, let ...
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If $|\int fg| \le M\|f\|_p$ for all $f\in L^p$, show that $g \in L^{q}$ and $\|g\|_q \le M$, where $1/p +1/q=1$

Let $g$ be an integrable function on $[0,1]$ and let $1 \leq p < \infty$. Suppose there is a constant $M$ such that $$ \left|\int f \;g \right| \leq M \; \|f\|_p $$ for all bounded measurable ...
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stuck on a problem about sequential weak* compactness of finite borel measures on a sigma compact metric space

I was working through a problem In a book on functional analysis. the problem is Let X be a $ \sigma $ compact metric space such that the space of bounded real continuous functions on X $ C_b(X) $ is ...
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Let $w$ be a positive continuous function for which $\int_0^1 w(x)dx = \int_0^1 x^2w(x)dx = 1$. Prove that $\int_0^1 xw(x)dx < 1$.

Let $w$ be a positive continuous function for which $$\int_0^1 w(x)dx = \int_0^1 x^2w(x)dx = 1.$$ Prove that $\int_0^1 xw(x)dx < 1$. I was thinking of using the Reisz Representation Theorem for ...
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Riesz representation theorem on Hilbert spaces - validation of proof

Is the following proof of Riesz representation theorem correct? I am following notation of Bachmann & Narici. Notation $\tilde{X}$: conjugate space of $X$ $(\cdot,\cdot)$: inner product $[S]$: ...
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Define a linear functional $T$ on $V$ by $Tv$ = $\langle v, u\rangle$. What is $T^∗ (\alpha)$ for a scalar $\alpha$ where $T^*$ is the adjoint.

Now I do understand that the question might involve using Riesz representation as it involves a linear functional and we know that it can be written using an inner product. So $u$ is the representer ...
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A counterexample of Riesz Representation Theorem?

I am working on the exercise 6.B.15 from Axler's book: Linear Algebra Done Right. This problem states that the Riesz Representation Theorem may fail on an infinite-dimensional vector space: Suppose $...
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Understanding proof of the Riesz Representation Theorem

I am studying Stanislaw Lojasiewicz book - "An introduction to the Theory of Real Functions" and I do not uderstand few things. I hope you'll help me. Here is what is written: G is an open set and $\...
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84 views

adjoint transformation intuition

I can't find the connection between the Riesz Representation Theorem and inner product spaces and the adjoint transformation. what I understood that dual spaces enables us to have an transpose ...
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65 views

What is the dual of the space $V=\{f: K\to \mathbb R^n, f(x)=Ax+b\}$ with $K\subset\mathbb R^n$ compact?

(1) Could anyone tell me how to find a Dual space of the following space of continuous functions of the following form? And dual maps? $V=\{f: K\to \mathbb R^n, f(x)=Ax+b\}$, where $K$ is a compact ...
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224 views

Riesz Representation Theorem geometric intuition

We just learned in our linear algebra class about the Riesz Representation Theorem, which states that if $V$ is finite-dimensional and $f$ is a linear functional on $V$, then there is a unique vector $...
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42 views

Prove that representations of G over F form a ring

I am trying to solve the question: Representations of a group G over a field F form a ring R. Am I supposed to use something like Grothendieck rings or I don't know?
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86 views

Representation of a bounded linear operator $T: c \to c$.

Let $T: c\to c$ be a bounded linear operator, where $c$ is the vector space of convergent real sequences. How can we prove that there exists an infinite matrix $A=(a_{n,k}: n,k\ge 1)$ such that $T(x)=...

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