Physlib.QuantumMechanics.DDimensions.SpaceDHilbertSpace.SchwartzSubmodule

The continuous linear map $\iota: \mathcal{S}(\text{Space } d, \mathbb{C}) \to L^2(\text{Space } d, \mathbb{C})$ embeds the Schwartz space $\mathcal{S}(\text{Space } d, \mathbb{C})$ of rapidly decreasing smooth functions into the Hilbert space $L^2(\text{Space } d, \mathbb{C})$ of square-integrable functions.

For a natural number $d$, this definition represents the subspace of the Hilbert space $L^2(\mathbb{R}^d, \mathbb{C})$ consisting of functions that are in the Schwartz space $\mathcal{S}(\mathbb{R}^d, \mathbb{C})$. Specifically, it is the range of the continuous linear inclusion map $\iota: \mathcal{S}(\mathbb{R}^d, \mathbb{C}) \to L^2(\mathbb{R}^d, \mathbb{C})$.

For any natural number $d$, this instance defines a coercion from the Schwartz submodule of the Hilbert space $L^2(\mathbb{R}^d, \mathbb{C})$ to the space of functions $\mathbb{R}^d \to \mathbb{C}$. This allows an element $\psi$ of the Schwartz submodule to be treated as a function, enabling the notation $\psi(x)$ for $x \in \mathbb{R}^d$.

For any natural number $d$, let $\psi$ be an element of the Schwartz subspace of the Hilbert space $L^2(\mathbb{R}^d, \mathbb{C})$. For any $x \in \mathbb{R}^d$, the value of the underlying $L^2$ function $\psi.\text{val}$ at $x$ is equal to the value of $\psi$ at $x$ when treated as a function through coercion, i.e., $\psi.\text{val}(x) = \psi(x)$.

For any natural number $d$, the Schwartz subspace $\mathcal{S}(\mathbb{R}^d, \mathbb{C})$ is dense in the Hilbert space $L^2(\mathbb{R}^d, \mathbb{C})$.

For a natural number $d$, this definition establishes the linear equivalence (isomorphism of $\mathbb{C}$-vector spaces) between the Schwartz space $\mathcal{S}(\mathbb{R}^d, \mathbb{C})$ and the corresponding subspace of the Hilbert space $L^2(\mathbb{R}^d, \mathbb{C})$ consisting of (the equivalence classes of) Schwartz functions. This is constructed by restricting the codomain of the continuous linear inclusion $\iota: \mathcal{S}(\mathbb{R}^d, \mathbb{C}) \to L^2(\mathbb{R}^d, \mathbb{C})$ to its range.

For any natural number $d$ and any Schwartz function $f \in \mathcal{S}(\mathbb{R}^d, \mathbb{C})$, the element of the Hilbert space $L^2(\mathbb{R}^d, \mathbb{C})$ corresponding to $f$ under the linear isomorphism $\text{schwartzEquiv}$ is equal to $f$ almost everywhere with respect to the Lebesgue measure.

For any natural number $d$ and Schwartz functions $f, g \in \mathcal{S}(\mathbb{R}^d, \mathbb{C})$, the inner product of their corresponding elements in the Hilbert space $L^2(\mathbb{R}^d, \mathbb{C})$ (under the linear isomorphism $\text{schwartzEquiv}$) is given by the integral of the product of the complex conjugate of $f$ and $g$: $$\langle \text{schwartzEquiv } f, \text{schwartzEquiv } g \rangle_{\mathbb{C}} = \int_{\mathbb{R}^d} \overline{f(x)} g(x) \, dx$$ where $\overline{f(x)}$ denotes the complex conjugate of $f(x)$.

For any natural number $d$ and Schwartz functions $f, g \in \mathcal{S}(\mathbb{R}^d, \mathbb{C})$, if the corresponding elements in the Hilbert space $L^2(\mathbb{R}^d, \mathbb{C})$ (under the linear isomorphism $\text{schwartzEquiv}$) are equal almost everywhere with respect to the volume measure, then $f$ and $g$ are equal as functions.

For any natural number $d$, the Schwartz submodule, which consists of the image of the Schwartz space $\mathcal{S}(\text{Space } d, \mathbb{C})$ under the inclusion into the Hilbert space $L^2(\text{Space } d, \mathbb{C})$, is dense in $L^2(\text{Space } d, \mathbb{C})$.

For a natural number $d$ and Schwartz functions $f, g \in \mathcal{S}(\text{Space } d, \mathbb{C})$, the inner product of their corresponding images in the Hilbert space $L^2(\text{Space } d, \mathbb{C})$ is equal to the integral over $\text{Space } d$ of the complex conjugate of $f(x)$ multiplied by $g(x)$: $$\langle \text{schwartzEquiv}(f), \text{schwartzEquiv}(g) \rangle_{\mathbb{C}} = \int_{\text{Space } d} \overline{f(x)} g(x) \, dx$$ where $\overline{f(x)}$ denotes the complex conjugate of $f(x)$ and the inner product is the standard $L^2$ inner product.

For any natural number $d$, the kernel of the continuous linear inclusion map $\iota: \mathcal{S}(\text{Space } d, \mathbb{C}) \to L^2(\text{Space } d, \mathbb{C})$ is the trivial submodule $\{0\}$.