Physlib.QuantumMechanics.OneDimension.HilbertSpace.SchwartzSubmodule

The continuous complex-linear map $\iota : \mathcal{S}(\mathbb{R}, \mathbb{C}) \to L^2(\mathbb{R}, \mathbb{C})$ that embeds the space of Schwartz functions $\mathcal{S}(\mathbb{R}, \mathbb{C})$ into the Hilbert space of square-integrable functions $L^2(\mathbb{R}, \mathbb{C})$ with respect to the Lebesgue measure.

The continuous linear map $\iota : \mathcal{S}(\mathbb{R}, \mathbb{C}) \to L^2(\mathbb{R}, \mathbb{C})$ that embeds the space of Schwartz functions $\mathcal{S}(\mathbb{R}, \mathbb{C})$ into the Hilbert space of square-integrable functions $L^2(\mathbb{R}, \mathbb{C})$ is injective.

Let $\mathcal{S}(\mathbb{R}, \mathbb{C})$ be the space of Schwartz functions and $\iota : \mathcal{S}(\mathbb{R}, \mathbb{C}) \to L^2(\mathbb{R}, \mathbb{C})$ be the continuous linear inclusion into the Hilbert space of square-integrable functions. For any Schwartz function $\psi \in \mathcal{S}(\mathbb{R}, \mathbb{C})$, its image $\iota(\psi)$ is equal to $\psi$ almost everywhere with respect to the Lebesgue measure.

Let $\mathcal{S}(\mathbb{R}, \mathbb{C})$ be the space of Schwartz functions and let $\iota : \mathcal{S}(\mathbb{R}, \mathbb{C}) \to L^2(\mathbb{R}, \mathbb{C})$ be the continuous linear inclusion into the Hilbert space of square-integrable functions. For any two Schwartz functions $\psi_1, \psi_2 \in \mathcal{S}(\mathbb{R}, \mathbb{C})$, the inner product of their images in the Hilbert space is given by the integral over the real line of the complex conjugate of $\psi_1$ multiplied by $\psi_2$: $$\langle \iota(\psi_1), \iota(\psi_2) \rangle_{\mathbb{C}} = \int_{-\infty}^{\infty} \overline{\psi_1(x)} \psi_2(x) \, dx$$