Physlib.Mathematics.InnerProductSpace.Submodule

Let $E$ and $F$ be inner product spaces over $\mathbb{C}$. Given a submodule $M \subseteq E \times F$, the definition `submoduleToLp M` is the submodule of $E \times F$ equipped with the $L^2$ norm (the Euclidean norm $\sqrt{\|e\|^2 + \|f\|^2}$), consisting of the same underlying elements as $M$. This reinterpretation allows the submodule to inherit the inner product structure of the $L^2$ product space rather than the default product space structure (which uses the supremum norm).

Let $E$ and $F$ be inner product spaces over $\mathbb{C}$, and let $M \subseteq E \times F$ be a submodule. For any element $f \in E \times F$, $f$ belongs to $M$ if and only if its reinterpretation in the $L^2$ product space (the product space $E \times F$ equipped with the Euclidean norm $\sqrt{\|e\|^2 + \|f\|^2}$), denoted by $\text{toLp}_2(f)$, belongs to the reinterpreted submodule $\text{submoduleToLp}(M)$.

Let $E$ and $F$ be inner product spaces over $\mathbb{C}$, and let $M$ be a submodule of the product space $E \times F$ (equipped with the default supremum norm topology). Let $\text{submoduleToLp}(M)$ denote the reinterpretation of $M$ as a submodule of the product space $E \times F$ equipped with the $L^2$ norm (the Euclidean norm $\|(e, f)\|_2 = \sqrt{\|e\|^2 + \|f\|^2}$). Then the reinterpretation of the topological closure of $M$ is equal to the topological closure of the reinterpreted submodule $\text{submoduleToLp}(M)$.

Let $E$ and $F$ be inner product spaces over $\mathbb{C}$, and let $M \subseteq E \times F$ be a submodule. For any element $f \in E \times F$, $f$ belongs to the topological closure $\overline{M}$ (with respect to the default product topology) if and only if its reinterpretation in the $L^2$ product space $E \times F$ (equipped with the Euclidean norm $\|(e, f)\|_2 = \sqrt{\|e\|^2 + \|f\|^2}$), denoted by $\text{toLp}_2(f)$, belongs to the topological closure of the reinterpreted submodule $\text{submoduleToLp}(M)$.

Let $E$ and $F$ be inner product spaces over $\mathbb{C}$, and let $M$ be a submodule of the product space $E \times F$. For any element $g = (g_1, g_2) \in F \times E$, $g$ belongs to the adjoint submodule $M.\text{adjoint}$ if and only if the transformed element $(g_2, -g_1)$, when viewed in the $L^2$ product space $E \times F$ (the product space equipped with the Euclidean norm $\sqrt{\|e\|^2 + \|f\|^2}$), belongs to the orthogonal complement of the reinterpreted submodule $\text{submoduleToLp } M$.

Let $E$ and $F$ be inner product spaces over $\mathbb{C}$, and let $M \subseteq E \times F$ be a submodule. For any element $f \in E \times F$, $f$ belongs to the double adjoint of $M$ (denoted $M.\text{adjoint.adjoint}$) if and only if its reinterpretation in the $L^2$ product space $E \times F$ (equipped with the Euclidean norm $\sqrt{\|e\|^2 + \|f\|^2}$), denoted by $\text{toLp}_2(f)$, belongs to the double orthogonal complement of the reinterpreted submodule, denoted by $(\text{submoduleToLp } M)^{\perp \perp}$.

Let $E$ and $F$ be inner product spaces over $\mathbb{C}$, and let $M \subseteq E \times F$ be a submodule. For any element $g = (g_1, g_2) \in F \times E$, $g$ belongs to the adjoint of the topological closure of $M$ (denoted $\overline{M}.\text{adjoint}$) if and only if the transformed element $(g_2, -g_1)$, when viewed in the $L^2$ product space $E \times F$ (equipped with the Euclidean norm $\|(e, f)\|_2 = \sqrt{\|e\|^2 + \|f\|^2}$), belongs to the orthogonal complement of the topological closure of the reinterpreted submodule, denoted $(\overline{\text{submoduleToLp } M})^\perp$.