Physlib.QuantumMechanics.HilbertSpaces.SpaceD.SchwartzSubmodule

For a natural number $d$ and a measure $\mu$, let the Schwartz submodule be the subspace of the Hilbert space $L^2(\text{Space } d, \mathbb{C})$ consisting of equivalence classes of Schwartz functions. This definition provides a coercion that allows an element $\psi$ of this submodule to be treated as a function from $\text{Space } d$ to $\mathbb{C}$, enabling the notation $\psi(x)$ for $x \in \text{Space } d$.

For any natural number $d$ and any Schwartz function $f \in \mathcal{S}(\text{Space } d, \mathbb{C})$, let $\text{schwartzEquiv}(f)$ be its image under the linear isomorphism between the Schwartz space and its corresponding submodule in the Hilbert space $L^2(\text{Space } d, \mathbb{C})$. The coercion of this image into the Hilbert space is equal to the image of $f$ under the continuous linear inclusion $\text{schwartzIncl} : \mathcal{S}(\text{Space } d, \mathbb{C}) \to L^2(\text{Space } d, \mathbb{C})$.

For any natural number $d$ and any Schwartz function $f \in \mathcal{S}(\mathbb{R}^d, \mathbb{C})$, let $\Phi(f)$ be the image of $f$ under the linear isomorphism `schwartzEquiv` into the Schwartz submodule of the Hilbert space $L^2(\mathbb{R}^d, \mathbb{C})$. Then $\Phi(f)$ is equal to $f$ almost everywhere with respect to the Lebesgue volume measure.

For any dimension $d \in \mathbb{N}$, let $\mathcal{S}(\mathbb{R}^d, \mathbb{C})$ be the Schwartz space and let $\mathcal{H}_{\mathcal{S}} \subseteq L^2(\mathbb{R}^d, \mathbb{C})$ be the submodule of the Hilbert space consisting of Schwartz functions. For any element $\psi \in \mathcal{H}_{\mathcal{S}}$, let $f = \text{schwartzEquiv}^{-1}(\psi)$ be its image in $\mathcal{S}(\mathbb{R}^d, \mathbb{C})$ under the inverse linear isomorphism. Then the function $f$ is equal to $\psi$ almost everywhere with respect to the Lebesgue measure.

Let $d$ be a natural number and let $\text{Space } d$ be the $d$-dimensional real inner product space (isomorphic to $\mathbb{R}^d$). For any two Schwartz functions $f, g \in \mathcal{S}(\text{Space } d, \mathbb{C})$, if their corresponding elements in the Hilbert space $L^2(\text{Space } d, \mathbb{C})$ are equal almost everywhere with respect to the Lebesgue measure, then $f = g$.

For a dimension $d = 0$, the Schwartz submodule of the Hilbert space $L^2(\text{Space } d, \mathbb{C})$ is equal to the entire Hilbert space.

Let $\text{Space } d$ be a $d$-dimensional normed additive commutative group (equivalent to $\mathbb{R}^d$) with a measure $\mu$. Let $L^2(\text{Space } d, \mu)$ be the Hilbert space of square-integrable functions, and let $\text{SchwartzSubmodule}(d, \mu)$ be its submodule consisting of $\mu$-almost everywhere equivalence classes of Schwartz maps. For a subset $\Omega \subseteq \text{Space } d$, let $L^2(\Omega, \mu)$ be the Hilbert space of square-integrable functions on $\Omega$, and let $P_\Omega: L^2(\text{Space } d, \mu) \to L^2(\Omega, \mu)$ be the linear projection map that restricts functions to $\Omega$. An element $\psi \in L^2(\Omega, \mu)$ belongs to the restricted Schwartz submodule $\text{SchwartzSubmoduleOn}(\Omega, \mu)$ if and only if there exists a global Schwartz class $\phi \in \text{SchwartzSubmodule}(d, \mu)$ such that $P_\Omega(\phi) = \psi$.