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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34 views

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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45 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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51 views

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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69 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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38 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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66 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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24 views

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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20 views

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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39 views

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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97 views

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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25 views

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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39 views

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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33 views

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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49 views

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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60 views

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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74 views

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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25 views

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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48 views

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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30 views

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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39 views

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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51 views

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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46 views

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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43 views

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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39 views

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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25 views

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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33 views

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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32 views

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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61 views

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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29 views

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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50 views

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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52 views

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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79 views

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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100 views

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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106 views

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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66 views

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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82 views

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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56 views

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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57 views

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
77 views

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
100 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 ...
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59 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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79 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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110 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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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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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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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 ...

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