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).
2
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1answer
44 views
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 ...
0
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1answer
37 views
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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0answers
12 views
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
$$
\int_{x \in ...
1
vote
1answer
28 views
Riesz Representation Theorem for Functional on Hilbert space
Let $H$ be a Hilbert space and $f \in H^*$. Then there is unique $y \in H$ such that $$ f(x)= \langle x,y \rangle$$ for all $x\in H. $
In the proof of this, first we use projection theorem and ...
2
votes
1answer
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\...
0
votes
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 ...
0
votes
1answer
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 ...
1
vote
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 ...
0
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0answers
12 views
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$}...
0
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0answers
47 views
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)=\...
1
vote
1answer
68 views
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 $...
1
vote
0answers
27 views
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 \; \...
3
votes
2answers
52 views
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 ...
4
votes
1answer
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 ...
2
votes
0answers
35 views
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 ...
0
votes
0answers
34 views
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) ...
1
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1answer
40 views
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 $\forall x,y,z \in H$, $\lambda, \mu \in \mathbb{C}$,
$$ \langle x, \...
0
votes
1answer
72 views
Suppose $g$ satisfies equation $G(f)= \displaystyle{\int fg d \mu}$ for all $f$ in $L_1$
Suppose $g$ satisfies equation $G(f)= \displaystyle{\int fg d \mu}$ for all $f$ in $L_1$ and that $c>1$. Let $E_c=\{ x: |g(x)|\geq c||G|| \}$, and define $f_c(x)$ to be $\pm1$ when $\pm|g(x)|\geq c|...
1
vote
1answer
127 views
Counterexample the riesz representation theorem
The exercises shows that the Riesz Representation does not hold on infinite-dimensional inner product spaces. I need help.
Suppose $C_{\mathbb{R}}([-1, 1])$ is the vector space of continuous real-...
0
votes
1answer
186 views
Walter Rudin Real and Complex Analysis Chapter 2
Walter Rudin Real and Complex Analysis Chapter 2 2.14 Riesz representation theorem the last step.
Why did he put the absolute value of $a$ ?
Is not it sufficient to assume $f$ is positive?
Proof.
...
0
votes
0answers
14 views
Parameter-independent constrained optimization approach
I try to understand how to rigorously use the Riesz theorem for the following problem, and I would very much appreciate if anybody could give me a hand on that. The following is definitely just my ...
1
vote
0answers
46 views
Applying Riesz Representation Theorem to prove $g \in L^{p}(E) = 0$ if integral of $fg$ is 0 for all $f$ in dense subset of $L^{q}(E)$
I know this question has been asked before, but I wanted to try a different proof and get help tying up a piece of the proof that wasn't clear to me in that answer for $p < \infty$ (basically it ...
0
votes
1answer
54 views
Uniqueness of Riesz Representation of $C^{*}[a,b]$
I have seen this statement: The dual space $C^{*}\left[a,b\right]$ of $C\left[a,b\right]$ is isometrically isomorphic to $BV_{0}\left[a,b\right]$.
where $$BV_{0}\left[a,b\right]=\left\{ \alpha\in BV\...
0
votes
0answers
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 ...
0
votes
1answer
54 views
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 $T\colon H\mapsto H$ ...
3
votes
0answers
43 views
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^...
2
votes
0answers
40 views
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^*...
4
votes
2answers
196 views
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 ...
6
votes
2answers
179 views
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 ...
2
votes
1answer
100 views
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 ...
2
votes
1answer
140 views
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 ...
1
vote
1answer
57 views
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(...
3
votes
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 ...
0
votes
1answer
66 views
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,...
0
votes
0answers
50 views
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 $dim(\mathcal{H})L$?
0
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1answer
96 views
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.
-3
votes
1answer
171 views
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 ...
1
vote
2answers
116 views
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 ...
3
votes
3answers
131 views
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 $X$...
1
vote
0answers
86 views
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 ...
0
votes
2answers
97 views
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 $x = (a_{1}, a_{2}, .......) \in l_{2}$, $$f(...
0
votes
1answer
42 views
Need a help in understanding theorem 6.1 in chapter 2 in Israel Gohberg.
The theorem and its proof is given in the following pictures:
But I could not understand:
$Q_{1}$ the line after equation(1), why $f_{i}(x)$ is given by the indicated form for all i?, and why it is ...
0
votes
1answer
117 views
Need a help in understanding example(1) on Riesz representation theorem.
The author said:
"A functional $F$ on $L^{2}([a,b])$ is bounded and linear iff there exists a $g \in L^{2}([a,b])$ such that $$F(f) = \int_{a}^{b} f(t) \bar{g}(t) dt,$$
for all $f \in L^{2}([a,b])$. ...
2
votes
1answer
92 views
Prove that : an operator $\ell$ satisfying, $\ell(f)\ge 0$ whenever $f\ge 0$ is bounded on $C_b(X)$ .
While reading some functional analysis note I came across the following theorem.
Riesz-Markov: (for linear forms on Wiener spaces) If $X$ is locally compact Hausdorff space and $\ell : C_b(X)\to \...
3
votes
1answer
168 views
Riesz representation theorem in $C^k([0,1])$
I'm trying to figure out the next exercise
Let be $k\geq 1$ and $C^k([0,1])$ the banach space of all k-times differentiable function with the norm
$$\lVert f\rVert_{k,\infty}=\sum_{i=0}^{k}\lVert ...
3
votes
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
votes
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 $\Lambda(f) = \int_{R^+} f \mathrm{d}\mu$ ...
1
vote
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
votes
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: ...
0
votes
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 $f_y: \mathcal H \to \mathbb K, f_y(x) := \langle x \,,\,y\rangle$...
