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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 nd Riesz- Markov theorem as well which extend to linear forms on Wiener spaces (space of continuous functions).

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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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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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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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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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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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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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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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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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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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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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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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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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Riesz Representation Theorem the last step in Real and Complex Analysis for Walter Rudin

I've read the proof of riesz representation theorem from (Real and Complex Analysis) for Walter Rudin with its 10 steps The last step which is the proof that this measure represents the functional. ...
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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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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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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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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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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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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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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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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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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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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 $$\...
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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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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 ...
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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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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 $f_y: \mathcal H \to \mathbb K, f_y(x) := \langle x \,,\,y\rangle$...
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Riesz representation for bounded function and continuous functional

I have found the following version of Riesz-Representation theorem: Let X be local compact hausdorff space. For any continuous functional $\psi $ on $C_0(X)$ (continuous function vanishing at ...
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Understanding Riesz–Markov–Kakutani representation theorem

The Riesz representation theorem is very easy to understand. Further, every continuous linear functional $A[f]$ over the space $C([0, 1])$ of continuous functions in the interval $[0,1]$ can be ...
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Aplication of Riesz representation theorem

For the purpose of practice I took on the following problem about the Riesz representation theorem which asks me to find the unique element whose existence the representation theorem ensures. Let $H ...
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Infinite product measure via Riesz representation theorem

I am curious about how we can get the existence of product measures in the infinite dimensional space by using the Riesz representation theorem. I am studying the Kolmogorov extension theorem by the ...
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Why is the space of finite Borel-measure dual to the space of finite continuous function.

Riesz-Representation-Theorem states that every positive linear functional $F$ for any finite continuous $f$ on a local compact space S one can find a unique borel measure, such that $$F(f)=\int f d\mu$...
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Riesz representation theorem giving a different result?

Evans p298 - Lax Milgram Theorem Let $H$ be a Hilbert space. Denote the inner product by $(\cdot,\cdot)$ and the natural dual pairing of spaces by $\langle\cdot,\cdot\rangle$. He gives us bilinear ...
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Philosophy behind Riesz–Markov–Kakutani representation theorem

The Riesz–Markov–Kakutani representation theorem tells us that, if $X$ is a locally-compact Hausdorff space, then monotone linear functionals $\mathcal{C}_{\mathrm{cs}}(X) \rightarrow \mathbb{R}$ are ...
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Find the vector in $\ell^2(\mathbb N)$ corresponding to $f$ as in the Riesz Representation Theorem

I've been trying to get my head around the Riesz Representation Theorem and I'm stuck on this question. Let $N \in \mathbb N$ and define $f: \ell^2(\mathbb N) \to \mathbb R$ by $f((a_n)_{n=1}^{\...