# 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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### Prove a Borel measure coincides with a Riesz measure on Borel $\sigma$-algebra

I try to prove the Lemma used here :Borel measure and Riesz measure To prove: If a Borel measure $\mu$ coincides with a Riesz measure $\lambda$ on any open set in $\mathbb{R^n}$, then they coincides ...
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### Problem $2.17$, Rudin's RCA (Dictionary Order Topology)

Problem $2.17$: Define the distance between the points $(x_1,y_1)$ and $(x_2,y_2)$ in the plane to be $$|y_1-y_2| \quad \text{if }x_1 = x_2, \quad\quad 1+|y_1 - y_2|\quad \text{if } x_1\ne x_2$$ Show ...
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### Compactness of set of measures with respect to weak-*topology

I am reading a proof of the Stone-Weierstrass theorem by De Branges, but I am having trouble understanding the following part. Let $E$ be a locally compact Hausdorff space and $C_0(E, \mathbb{C})$ the ...
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### If $V \hookrightarrow H$ are Hilbert spaces and $f \in H'$ then $f \in V'$?

Let $H=(H, (\cdot, \cdot)_H)$ and $V=(V, (\cdot, \cdot)_V)$ be Hilbert spaces such that $V \hookrightarrow H$ that is $V$ is continuously embedding in $H$. By Riesz representation theorem we can ...
1answer
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### A Banach space isomorphic to his dual is an Hilbert space? [duplicate]

By the Riesz representation theorem, we know that the Hilbert space $\mathcal{H}$ is isomorphic to his dual. Is the converse true ? Does the fact that a Banach space $E$ is isomorphic to his dual ...
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### What is the predual space of $H_0^1(\Omega)$?

I am trying to understand the predual space $X$ of $H_0^1(\Omega)$. My idea was to identify the predual space by the canonical embedding $i:X\to (H_0^1(\Omega))^{*}$. I know that a Hilbert space is ...
1answer
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### Riesz Representation Theorem, true for pre-Hilbert spaces and any functional?

I have some doubts about the Riesz theorem. Firstly can you check my proof? Fa = for all Fa (H,<,>) a pre Hilbert space Fa x in H different from the zero vector Fa F a functional on H Fa T: (H-&...
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### About a point in the the answer to “Riesz representation and vector-valued functions”.

In the Q&A Riesz representation and vector-valued functions, @anon gave an answer. why $m_{\phi}$ is well-defined? Could some one give some reference or explain more about it?
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### An alternative proof for Riesz Representation Theorem for $C([0,1])$

During my class in real analysis, my teacher mentioned an alternative method to prove Riesz's theorem for $C([0,1])$. He just mentioned the method, but did not prove it. (A usual way of doing this is ...
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### Proving that a Hilbert space $\mathcal{H}$ is isometrically isomorphism to $l_2(I)$.

Let $\mathcal{H}$ an arbitrary Hilbert space no necessary Separable. Let $\{u_i:i\in I\}$ a orthonormal basis of $\mathcal{H}$ where $I$ is a uncountable set. Let $\mathbb{K}$ a field and $l_2(I)$ the ...
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### Existence and uniqueness of the adjoint of a linear map.

I have this proposition from Karlheinz Spindler's Abstract algebra with applications vol. 1, page 303. It is proved there, but I tried it by myself based on that prove. I would like to know if my ...
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### Confusion about uniqueness of $\sigma$-algebra in Riesz-representation theorem

In Rudin's "real analysis and complex analysis" the Riesz representation theorem is proved and used to define the lebesgue measure on $\mathbb{R}^n$. I have a question about this. First, let ...
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### If $|\int fg| \le M\|f\|_p$ for all $f\in L^p$, show that $g \in L^{q}$ and $\|g\|_q \le M$, where $1/p +1/q=1$

Let $g$ be an integrable function on $[0,1]$ and let $1 \leq p < \infty$. Suppose there is a constant $M$ such that $$\left|\int f \;g \right| \leq M \; \|f\|_p$$ for all bounded measurable ...
1answer
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### stuck on a problem about sequential weak* compactness of finite borel measures on a sigma compact metric space

I was working through a problem In a book on functional analysis. the problem is Let X be a $\sigma$ compact metric space such that the space of bounded real continuous functions on X $C_b(X)$ is ...
1answer
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### Let $w$ be a positive continuous function for which $\int_0^1 w(x)dx = \int_0^1 x^2w(x)dx = 1$. Prove that $\int_0^1 xw(x)dx < 1$.

Let $w$ be a positive continuous function for which $$\int_0^1 w(x)dx = \int_0^1 x^2w(x)dx = 1.$$ Prove that $\int_0^1 xw(x)dx < 1$. I was thinking of using the Reisz Representation Theorem for ...
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### Riesz representation theorem on Hilbert spaces - validation of proof

Is the following proof of Riesz representation theorem correct? I am following notation of Bachmann & Narici. Notation $\tilde{X}$: conjugate space of $X$ $(\cdot,\cdot)$: inner product $[S]$: ...
1answer
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### Define a linear functional $T$ on $V$ by $Tv$ = $\langle v, u\rangle$. What is $T^∗ (\alpha)$ for a scalar $\alpha$ where $T^*$ is the adjoint.

Now I do understand that the question might involve using Riesz representation as it involves a linear functional and we know that it can be written using an inner product. So $u$ is the representer ...
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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 ...
1answer
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### What is the dual of the space $V=\{f: K\to \mathbb R^n, f(x)=Ax+b\}$ with $K\subset\mathbb R^n$ compact?

(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 ...
1answer
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