# 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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### 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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### 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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### 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 ...