Question

Let $H$ be the Hilbert space of square-integrable functions on the measure space of reals. Define $H \xrightarrow{s} \mathbf{C}$ by $\varsigma(\Omega)=\int_{\mathbf{R}} f e^{i k x} d \mu$, where $k$ is any number. This $\varsigma$ is apparently continuous and linear, and apparently not of the form $\varsigma(h)=(\underline{h}, h)$. What is wrong?

   Let $H$ be the Hilbert space of square-integrable functions on the measure space of reals. Define $H \xrightarrow{s} \mathbf{C}$ by $\varsigma(\Omega)=\int_{\mathbf{R}} f e^{i k x} d \mu$, where $k$ is any number. This $\varsigma$ is apparently continuous and linear, and apparently not of the form $\varsigma(h)=(\underline{h}, h)$. What is wrong?

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Mathematical physics

Mathematical physics

Robert Geroch 1st Edition

Chapter 48, Problem 330

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We are given a Hilbert space \( H \) of square-integrable functions on the real line, denoted as \( L^2(\mathbb{R}) \). The function \( \varsigma: H \to \mathbb{C} \) is defined by \( \varsigma(f) = \int_{\mathbb{R}} f(x) e^{ikx} \, d\mu(x) \), where \( k \) is a Show more…

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Let $H$ be the Hilbert space of square-integrable functions on the measure space of reals. Define $H \xrightarrow{s} \mathbf{C}$ by $\varsigma(\Omega)=\int_{\mathbf{R}} f e^{i k x} d \mu$, where $k$ is any number. This $\varsigma$ is apparently continuous and linear, and apparently not of the form $\varsigma(h)=(\underline{h}, h)$. What is wrong?

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