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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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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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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dual space, dual map
(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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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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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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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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Stein's interpolation theorem (analytic interpolation)
Let $G^{\alpha}_k$ with $\alpha\in\Bbb C$ et $k\in\Bbb N^*$ be analytic family of operators such that:
For $\tau\in\Bbb R$
$$\|G^{i\tau}_k f\|_{\infty}\leq C(1+|\tau|)^{\frac{1}{2}}\|f\|_1 $$
and ...
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Defining borel measure on unit circle in complex plane (Chapter 4, section 23 RCA Rudin)
I am going through the section on trigonometric series, chapter 4 of RCA Rudin, where he defines $L^p$ norm of functions defined on the unit circle. However a development of measure on the unit circle ...
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Meaning of Riesz representations in a layman's term?
I have read Riesz representation theorem quite a lot of times. While understanding the mathematical meaning, that is, for each element $x$ in a Hilbert space $X$, there will be a linear and bounded ...
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Rudin, Riesz Representation Step X: Why do we need $|a|$?
My question has to do with the inequalities at the end. However, I will summarize the step to give some context to my question.
We want to show that for some complex functional $\Lambda $on $C_c(X) $ ...
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show that for every $y$ the sequence $(\|\phi_{x_{n}}(y) \|)_{n \in \mathbb{N}}$ is bounded.
Here is the question and its solution (a solution given to me by Keefer):
Let $\{x_{n}\}$ be an unbounded sequence in Hilbert $\mathcal{H}.$ Prove that there exists a vector $x \in \mathcal{H}$ such ...
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Sequence of continuous fuctions with compact support converges to 1.
Let $X\subset\mathbb{R}^d$ be an unbounded closed set and $C_0$ is the space of all continuous functions $h: X\to \mathbb{R}$ with compact support. I'm searching for the sequence $\{ h_n \} \subset ...
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Intuition for Reisz Representation Theorem
I'm currently working on understanding The Reisz Representation Function in Papa Rudin. I'm wondering why we choose as our seed of a measure to be
$$\hspace{-2in} (1) \hspace{2in} \mu(V) = \sup\{ \...
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Dual space of the Intersection of locally convex vector spaces
Let $S \neq \emptyset$ and let $\big((E_s,\mathcal{T}_s)\big)_{s \in S}$ be a family of locally convex vector subspaces of the same vector space. Denote by $E_s^*$ the dual space of $(E_s,\mathcal{T}...
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1answer
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Dimension of a closed subspace of $C[0,1]$
Let $M$ be a closed subspace of $L^2([0, 1]; m)$ that is contained in $C([0, 1])$, where $m$ denotes Lebesgue measure. I have already proved that there exists some positive number $K$ such that $\|f\|...
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Dual Space of General $L^p$ Space Which Takes Values in Banach Space
Last week, our functional analysis course covered Riesz Representation theorem for $L^p(X,\mu),(1\leq p < \infty)$, namely, $(L^p(X,\mu))^* = L^q(X,\mu)$. And I was stuck with this homework problem ...
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Riesz representation for a countable family of functions
In reading this I have found the following result that I don't know how to prove precisely:
Theorem: Let $(X,d)$ be a compact metric space and let $C(X)$ be the Banach space of real valued ...
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Help with Proving This Lemma
Lemma Let $\Omega$ be a $\sigma$-finite measure space and $J \colon \Omega \to [0, \infty]$ a measurable function. If $1 < p < \infty$ and $F \ge 0$ then the following statements are equivalent:
...
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Dual space of $C^{1,2}_b([0, T]\times O)$
Let $T>0$ and $O\subseteq \mathbb R$ an open set. Let $C^{1,2}_b([0, T]\times O)$ be the space of functions $u$ continuous bounded such that $\partial_t u$, $\partial_x u$ and $\partial_{xx}u$ are ...
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Proving the Riesz Representation Theorem for the space of 1 point with the counting measure.
I am trying to solve this question:
Give an example of a measure space $(X, \mathfrak{M}, \mu)$ for which the Riesz Representation Theorem does extend to the case $p=\infty.$
My Trial:
I am trying ...
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Understanding a step in the proof of RRT for the dual of $L^p(X,\mu)$ on pg.402.(Royden “Real Analysis” 4th edition )
Here is the step that I do not understand:
Should not the equality before the last contains a power of $p$ over the sum and an integral sign according to the definition of p-norm , or maybe I do ...
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Dual space of continuous functions vanishing at infinity. Euclidean version.
Hi! Seeing Grafakos theorem 2.5.8, it is said that the finite borel measures space is dual of the space of continuous functions that tends to zero at infinity. To see this, I searched Conway for that ...
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Connection between Representer Theorem and Mercer's Theorem?
In my studies about Reproducing Kernel Hilbert Spaces (RKHS) I came across a bit of confusion as to what kind of functions are defined in this RKHS $\mathcal{H}$. Specifically, the Representer Theorem ...
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Riesz Representation Theorem Positivity Question
In $\mathbb{R}^S$, where $S < \infty$, if there is a linear functional such that $F(x) = 0$, $\forall x \in M$ and $F(y) > 0$, $\forall k \in K$ where $K$ and $M$ are linear subspaces, then can ...
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Application of Hahn-Banach and Riesz Representation Theorems on Inner Product Space of Continuously Differentiable Functions
The following is a problem from a practice exam I was given for a course I am currently in:
Let $X$ be the space of all continuously differentiable functions $f : [0,1] → \Bbb{K}$ with $f(0) = 0$, ...
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Riesz Representation Theorem in a compact space
What simplifications would be found going through the proof of the Riesz Representation Theorem assuming the space X to be compact (or even compact metric) rather than just locally compact?
(I'm ...
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Riesz Lemma and isomorphism
The Reisz' Lemma reads
Let $H$ be a hilbert space over a field $\mathbb{K}$. Then,$$\forall f \in H^{*} \quad \exists! y \in H: f(x)=\langle x,y \rangle \, \forall x \in H , \|f\|=\|y\|$$
Now, let $\...
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1answer
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Proving a Markov chain has a stationary measure
I'm in a measure-theoretic probability class and I am studying for an upcoming exam. Here is a problem from a book that I am studying from for this exam. But I am really not sure how to solve this ...
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Is there any name for the property denoted by $K \prec f$ or $f \prec V$, where $K, V$ are sets and $f$ is a function?
I'm studying the proof of the Riesz representation theorem for regular measures using Real and Complex Analysis of Walter Rudin. In page 38, he introduces some notation that reads:
Notation:
We ...
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1answer
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Using Riesz Representation Theorem to find positive linear functionals
I am having trouble using the Riesz representation theorem in concrete examples. For instance, I came upon this problem (The RRT is the one in Chapter 2 of Papa Rudin and I preserve his notation):
...
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Rudin's Riesz Representation Theorem assuming X is a compact space
In Rudin's book Real and Complex Analysis there's an exercise in Chapter 2, the number 19, that challenges you to make some simplifications in Riesz Representation Theorem's proof assuming the space ...
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Existence and uniqueness of the adjoint of a linear operator between Hilbert spaces
Let $H_i$ be a $\mathbb C$-Hilbert space, $T$ be a linear operator from $H_1\to H_2$ and $T^\ast$ be a linear operator from $H_2$ to $H_1$ with $$\langle T^\ast y,x\rangle_{H_1}=\langle y,Tx\rangle_{...
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Is the function of a continuous linear functional to its kernel continuous?
Let $\mathcal{H}$ be a Hilbert space, $L\in \mathcal{H}^*$ be a bounded continuous linear functional. Suppose that $\ker\:L \neq \mathcal{H}$.
We know that the dimension of $\ker \:L$ is $1$, i.e. ...
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Riesz representation theorem for Bochner integral
Suppose $X$ and $W$ are Hilbert space. Let $F:X\to W$ be a vector-valued function which is Bochner integrable. Define functional $L:W\to\mathbb{C}$ as follows
$$L(f):=\int_X\langle f,F(x)\rangle_W d\...
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Find the Riesz representative of F using the orthogonal set to (ker(F))
Let F be a linear function over the real field such that F(1,0,0)=2 , F(0,1,0)=1 and F(0,0,1)=-1, find an orthogonal set to ker(F) and us it to calculate the Riesz representative of F
Is it necessary ...
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A question about the surjectivity of an operator in the proof of the Riesz representation theorem
I have a question from Brezis' Functional Analysis- Sobolev Spaces and Partial Differential Equations.
In the proof of Thm 4.11 (Riesz representation theorem) for $L^p$ spaces ($1 < p < \infty$)...
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Riesz-Markov-Kakutani Theorem: Total variation Norm and weak*
Let $X$ be a compact Hausdorff space, the Riesz-Markov-Kakutani theorem states that the topological dual of $C(X)$ is the space $M(X)$ of regular countably additive complex Borel measure on $X$ ...
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1answer
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Existence of adjoint via Riesz Representation Theorem
In Linear Algebra Done Right we have a theorem that states
Riesz Representation Theorem : Suppose $V$ is finite-dimensional and $A$ is a linear functional on $V$. Then there is a unique vector $u$ ...
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Proof Correct? Show that $\sum_{n \in \mathbb N}s_{n}t_{n}<\infty \Rightarrow s \in \ell^{1}$
Let $(s_{n})_{n}\subseteq \mathbb R$ and $c_{0}$ the space of null sequences. Show that $\sum_{n \in \mathbb N}s_{n}t_{n}<\infty \operatorname{for all }t\in c_{0} \Rightarrow s \in \ell^{1}$.
My ...
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Radon-Nikodym derivative of a finitely additive measure
I came across the following question asked in a prelims of UW.
Let $(X,F,\mu)$ be a finite measure space, and $\lambda$ be a finitely-additive, non-negative, real valued set function on $F$ such ...
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1answer
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Why is boundedness sufficient for well-definedness
Let $p, q \in {]1,\infty[}$ where $\frac{1}{p}+\frac{1}{q}=1$ and define
$J\colon L^{q} \to (L^{p})^{*}, f \mapsto \ell_{f}: L^{p} \ni g\mapsto \ell_{f}(g)=\int_{X}fg\,d\mu$
My question is related ...
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Using Riesz Representation Theorem to show a set of functionals is dense in H*
I am currently struggling with the following question:
Let $H$ be a Hilbert space with basis $\mathcal{E}$ and $φ_e (h)=〈h,e⟩$. Show that the set of functionals, span{$φ_e:e\in \mathcal{E}$}, is ...
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2answers
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Riesz representation for products?
Given is a continuous linear functional $T:C_c^0(\mathbb{R})\otimes C_c^0(\mathbb{R}) \rightarrow \mathbb{R}$ where $C_c^0(\mathbb{R})$ is the space of continuos functions with compact support. Since $...
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1answer
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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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1answer
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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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1answer
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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
41 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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1answer
55 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
69 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 ...
