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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 Representation Theorem measure is absolutely continuos w.r.t. the Lebesgue measure?

Riesz representation theorem: Let $X$ be a locally compact Hausdorff space, and let $\Lambda$ be a positive linear functional on $C_c(X)$. There exists a $\sigma$-algebra $\mathfrak{M}$ in $X$ which ...
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Find a unique vector $y$ such that $g(x)=<x,y> $ for all $x \in V$ (Riesz Representation Theorem example)

Consider $g : M (R)_{2x2}$ → $R$ given by $g(A)=a_{11} + 2a_{12} + 3a_{32} +4a_{22}$. We consider on $M_{2x2} (R)$ the inner product given by $<A,B> = tr(A^t ,B)$. Find the vector $y$. I only ...
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Representation of vector-integral

Suppose that $H$ is a separable Hilbert space, $(X,\Sigma,\mu)$ be a finite measure space and let $L^2(\Sigma,H)$ denote the set of Borel measurable functions from $X$ to $H$ satisfying $$ \int_{x \in ...
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1answer
28 views

Riesz Representation Theorem for Functional on Hilbert space

Let $H$ be a Hilbert space and $f \in H^*$. Then there is unique $y \in H$ such that $$ f(x)= \langle x,y \rangle$$ for all $x\in H. $ In the proof of this, first we use projection theorem and ...
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38 views

Pre-Hilbert spaces and the Riesz Representation Theorem.

I'm looking for an example of a Hilbert space $(H,\langle \cdot,\cdot\rangle)$ that satisfies the following: In $H$ there exists an element $a$ such that $(H \backslash \{a\},\langle \cdot,\cdot\...
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1answer
33 views

Riesz representation theorem: Does the order matter?

Let $X$ be a Hilbert space. $J:X\rightarrow X',\hspace{1cm}J(x):=(\cdot,x)$ is a complex conjugated isometric isomorphism between $X$ and it's dual space $X'$. Would there be any problems as a ...
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32 views

Riesz representation theorem: show isometry

Let $X$ be a Hilbert space and $J:X\rightarrow X',J(X):=(\cdot,x)$ where $X'$ is the dual space of $X$. I have to show that $\|J(x)\|_{\sup}=\|x\|$. ''$\leq$'' is clear by the Cauchy-Schwarz ...
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1answer
32 views

An application of Riesz-Fréchet representation theorem

Given $C=\{e_1^*,e_2^*,...,e_n^*\}$, the canonical basis of $(R_n)^*$ (dual of $R_n$). I need to prove that: $\forall i \in I_n=\{1,...,n\}, \forall u \in R^n: <e_i^*,u>=e_i^tu$ I can use the ...
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Open set with compact closure has finite Radon premeasure

Let $X$ be a locally compact Hausdorff space and $I$ a positive linear functional on $C_{c}(X)$. For each nonempty open subset $O$ of $X$, define $μ(O) = sup${$ I(f): f∈C_{c}(X), 0≤f≤1, supp f ⊆ O$}...
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About Riesz representation theorem

Let $X=l_p^{(3)}$, where $1\lt p \lt \infty$, and $\phi(x) = x_1-2x_2+3x_3$. Decide whether $\phi$ is bounded, and if so, find $||\phi||$. So by marking $y=(1,-2,3)$, we can see that $\phi(x)=\...
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Show that $x_0 \in \overline{\langle M \rangle } $ if and only if $f(x_0) = 0, \; \forall f \in X^* : f|_M = 0$.

Exercise : Let $X$ be a normed space and $M \subset X$. Show that an element $x_0 \in X$ belongs to the set $\overline{\langle M \rangle}$ if and only if $f(x_0) = 0$ for all $f \in X^*$ such that $...
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Showing that $\exists f \in X^*$ : $\|f\| = \frac{1}{d(x_0,Y)}, \; f(x_0) = 1$ and $f(y) = 0$.

Exercise : Let $X$ be a normed space and $Y$ be a proper closed subspace of $X$. If $x_0 \notin Y$, show that there exists $f \in X^*$ such that : $$\|f\| = \frac{1}{d(x_0,Y)}, \; f(x_0) = 1 \; \...
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Proving that $\exists f \in X^*$ : $f(x) = \|x\|^2$ and $\|f\| = \|x\|$

Exercise : Let $X$ be a normed space. Prove that for all $x \in X$ there exists $f \in X^*$, such that $f(x) = \|x\|^2$ and $ \|f\| = \|x \|$. Thoughts : I apologise for not providing a proper ...
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171 views

Does Radon-Nikodym imply Riesz Representation Theorem?

In Axler's Linear Algebra Done Right we have the theorem 6.42: (Riesz Representation Theorem) Suppose $V$ is a finite dimensional inner product space and $\phi$ is a linear functional on $V$. Then ...
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Map between positive linear functionals over locally compact Hausdorff space to measure is 1-1?

$X$ is locally compact hausdorff space. Let $\Gamma$ denote the set of positive linear functional on $C_c(X)$(i.e. all continuous functions over $X$ with compact support.) Denote $M(X)$ set of Borel ...
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How to show this step of the Riez representation theorem

I am studying for Bartle's book the elements of integration and lebesgue measure. He leaves this passage as an exercise which is to show that there exists $ g \in L_q $ such that it is worth (8.10) ...
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Question about Bounded Hilbert Operator and Riesz Theorem

I was dealing with this: Let \begin{equation} [ \cdot, \cdot ] : H \times H \rightarrow \mathbb{C} \end{equation} such that $\forall x,y,z \in H$, $\lambda, \mu \in \mathbb{C}$, $$ \langle x, \...
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Suppose $g$ satisfies equation $G(f)= \displaystyle{\int fg d \mu}$ for all $f$ in $L_1$

Suppose $g$ satisfies equation $G(f)= \displaystyle{\int fg d \mu}$ for all $f$ in $L_1$ and that $c>1$. Let $E_c=\{ x: |g(x)|\geq c||G|| \}$, and define $f_c(x)$ to be $\pm1$ when $\pm|g(x)|\geq c|...
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1answer
127 views

Counterexample the riesz representation theorem

The exercises shows that the Riesz Representation does not hold on infinite-dimensional inner product spaces. I need help. Suppose $C_{\mathbb{R}}([-1, 1])$ is the vector space of continuous real-...
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186 views

Walter Rudin Real and Complex Analysis Chapter 2

Walter Rudin Real and Complex Analysis Chapter 2 2.14 Riesz representation theorem the last step. Why did he put the absolute value of $a$ ? Is not it sufficient to assume $f$ is positive? Proof. ...
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Parameter-independent constrained optimization approach

I try to understand how to rigorously use the Riesz theorem for the following problem, and I would very much appreciate if anybody could give me a hand on that. The following is definitely just my ...
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Applying Riesz Representation Theorem to prove $g \in L^{p}(E) = 0$ if integral of $fg$ is 0 for all $f$ in dense subset of $L^{q}(E)$

I know this question has been asked before, but I wanted to try a different proof and get help tying up a piece of the proof that wasn't clear to me in that answer for $p < \infty$ (basically it ...
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1answer
54 views

Uniqueness of Riesz Representation of $C^{*}[a,b]$

I have seen this statement: The dual space $C^{*}\left[a,b\right]$ of $C\left[a,b\right]$ is isometrically isomorphic to $BV_{0}\left[a,b\right]$. where $$BV_{0}\left[a,b\right]=\left\{ \alpha\in BV\...
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34 views

Rudin, Riesz Repr. Thm. Step II, argument for the limit

Im reading Rudin, Real and Complex Analysis, and I have a problem with convincing myself of the correctness of the derivation of the first inequality in step II of Riesz Representation Theorem in ...
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1answer
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Find a Unique Bounded Linear Mapping in the Hilbert Space

this is a question on Banach Spaces that I have encountered. I am new to this type of stuff, and despite doing many exercise questions, I am still unsure where to begin. Let $T\colon H\mapsto H$ ...
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Riesz representation (Royden, 4ed)

I'm reading Royden's Real Analysis 4ed, and have doubts about the proof of Theorem 5 in Section 8.1. The theorem says: if $T$ is a bounded functional on $L^p[a, b]$, then there is a function $g \in L^...
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Special Case of the Riesz Representation Theorem

I'm not sure if my solution for the following problem is valid. Let $(V, ( , ))$ be a finite-dimensional inner product space over $F$ and let $V^*$ denote its dual space. Prove that for all $l \in V^*...
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Riesz representation theorem - yet another “counter example”

this is a follow up on a previous question I asked. I was looking for examples of when the Riesz representation theorem doesn't hold because not all conditions are met. Meaning, I was looking for ...
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Examples of when the Riesz representation theorem doesn't hold

I was wondering if anyone could give me some interesting "counter examples" to the Riesz representation theorem about functionals over Hilbert spaces. When I say counter examples, I'm obviously ...
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1answer
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Riesz isomorphism and dual map

Let $V=\mathbf{R}[X]_{\leqslant 1}$ be equipped with inner product $\langle f,g\rangle=\int_{[-1,1]} f(x)x^2 g(x)\,dx$. Let $J:V\to V^*:u\mapsto\ell_u$ where $\ell_u(x)=\langle u,x\rangle$ be the ...
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1answer
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Proving the Riesz Representation Theorem for $\ell^p$.

State and prove a Riesz Representation Theorem for the bounded linear functionals on $\ell^p$, for $1 \le p < \infty$. Let $T : \ell^p \to \Bbb{R}$ be a bounded linear functional, and let $q$ be ...
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1answer
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Question in Proof of Riesz Representation Theorem

I'm trying to give a step-by-step proof of the Riesz Representation Theorem for the Dual of $L^{1}(X,\mu)$ and I've hit a wall trying to show $T:L^{\infty}(X,\mu)\to (L^{1}(X,\mu))^{*}$ defined by $T(...
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1answer
98 views

Dual of $L^p$ space avoiding reflexivity and Radon-Nikodym Theorem

Let $(X,\mathscr{F},\mu)$ be a measure space, with $\mu(X)<+\infty$, let $p\in [1,+\infty[$ and $q$ its Hölder-conjugate (that is, $1/p+1/q=1$). If $T\in (L^p(X,d\mu))^*$ is a continuous functional ...
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1answer
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Prove the original Riesz Representation using the bilinear form

In the linear space $R^n$ over $R$, we have an inner product which is the usual one. The problem is to use the following theorem to prove the Riesz Representation Theorem: For any bilinear form $B(x,...
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Equivalence of representations

Let $\pi$ be a finite dimensional unitary representation on a hilbert space $\mathcal{H}$. Then why is the left regular representation $L$ on $L^2(G,\mathcal{H})$ equivalent to $dim(\mathcal{H})L$?
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Riez representation theorem does not hold on infinite-dimensional vector spaces example [closed]

Show that the Riesz Representation Theorem does not hold on infinite-dimensional vector spaces without any hypotheses on the vector space V and linear functional.
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Rudin's Proof of the Riesz Representation Theorem [closed]

I am having trouble understanding step III in Rudin's proof of the Riesz Representation theorem: I'm not going to define any of the symbols that occur in the picture or provided any of the background ...
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2answers
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Isometry in Riesz's representation theorem

Riesz's representation theorem states that if $H$ is a Hilbert space and $H^*$ its dual space, then the map $\Phi$ which maps $x\in H$ to $x^*\in H^*, x^*y:=\langle x, y\rangle \,\forall\,y\in H$ is ...
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3answers
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A problem about the inverse of Riesz's representation theorem

Let $X$ be an inner product space. For any bounded linear functional $f$ on X, there exists a unique $x_f \in X$ s.t. for any $x \in X$, $f(x)=\langle x, x_f\rangle$, and $\|f\|=\|x_f\|$. Show that $X$...
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0answers
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Derivative of gradient

Consider $H$ some euclidean Hilbert space, a $C^2$ function $f\colon H\to\mathbb{R}$, and $x,y\in H$. In the context of some problem about convexity and positive definiteness of the second derivative ...
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2answers
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Need a help in the proof of an example of Riesz representation theorem.

The example said:"A functional $f$ on $l_{2}$ is linear and bounded iff there exists a $y = (\beta_{1}, \beta_{2}, .......) \in l_{2}$ such that for all $x = (a_{1}, a_{2}, .......) \in l_{2}$, $$f(...
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1answer
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Need a help in understanding theorem 6.1 in chapter 2 in Israel Gohberg.

The theorem and its proof is given in the following pictures: But I could not understand: $Q_{1}$ the line after equation(1), why $f_{i}(x)$ is given by the indicated form for all i?, and why it is ...
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1answer
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Need a help in understanding example(1) on Riesz representation theorem.

The author said: "A functional $F$ on $L^{2}([a,b])$ is bounded and linear iff there exists a $g \in L^{2}([a,b])$ such that $$F(f) = \int_{a}^{b} f(t) \bar{g}(t) dt,$$ for all $f \in L^{2}([a,b])$. ...
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1answer
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Prove that : an operator $\ell$ satisfying, $\ell(f)\ge 0$ whenever $f\ge 0$ is bounded on $C_b(X)$ .

While reading some functional analysis note I came across the following theorem. Riesz-Markov: (for linear forms on Wiener spaces) If $X$ is locally compact Hausdorff space and $\ell : C_b(X)\to \...
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1answer
168 views

Riesz representation theorem in $C^k([0,1])$

I'm trying to figure out the next exercise Let be $k\geq 1$ and $C^k([0,1])$ the banach space of all k-times differentiable function with the norm $$\lVert f\rVert_{k,\infty}=\sum_{i=0}^{k}\lVert ...
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0answers
394 views

Original Proof of Riesz Representation Theorem for $C([0,1])^*$

It is well known that Riesz Representation Theorem states that every positive linear functional $\Psi$ on $C_c(X),$ where $X$ is a locally compact Hausdorff space, can be realize as integration $$\...
3
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1answer
93 views

Riesz Representation Theorem on noncompact space

I meet this problem: $\Lambda(f)$ is a nonnegative bounded linear functional on $C[0,\infty)$. Assume $\Lambda(1) = 1$. Then $\Lambda$ has a representation $\Lambda(f) = \int_{R^+} f \mathrm{d}\mu$ ...
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2answers
75 views

Help with the step II in the proof of Riesz Representation Theorem in big Rudin

In Riesz Representation Theorem, $X$ is a locally compact Hausdorff space, and $\Lambda$ is a positive linear functional on $C_c(X)$ which is the set of all continuous functions on $X$ with compact ...
2
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0answers
63 views

Representation theorem for nonlinear functionals

Is there some analogon of the Riesz Representation theorem for nonlinear functionals on $L^p$ spaces? I would expect something like: A smooth (e.g. in the Fréchet sense) nonlinear functional $\Psi: ...
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1answer
65 views

Question concering a proof of the Riesz Representation Theorem

Riesz Representation Theorem states: For each $f \in \mathcal H^*$ there exists a unique $y \in \mathcal H$ such that $f = f_y$, where $f_y: \mathcal H \to \mathbb K, f_y(x) := \langle x \,,\,y\rangle$...