Physlib.QuantumMechanics.HilbertSpaces.TensorProducts.CompleteTensorProduct

Given a field $\mathbb{K}$ (which is either $\mathbb{R}$ or $\mathbb{C}$) and two inner product spaces $E$ and $F$ over $\mathbb{K}$, the complete tensor product $E \otimes_h F$ (also denoted $E \otimes_{h, \mathbb{K}} F$) is the Cauchy completion of the algebraic tensor product $E \otimes_{\mathbb{K}} F$ with respect to the metric induced by its natural inner product. This construction yields a Hilbert space.

This is the infix notation $E \otimes_h F$ for the complete tensor product (also known as the Hilbert tensor product) of two inner product spaces $E$ and $F$. It denotes the completion of the algebraic tensor product $E \otimes_{\mathbb{k}} F$ with respect to the norm induced by the tensor product's inner product.

For inner product spaces $E$ and $F$ over a field $\mathbb{k}$, the notation $E \otimes_{\text{h}}[\mathbb{k}] F$ denotes the complete tensor product (also known as the Hilbert tensor product). This is defined as the completion of the algebraic tensor product $E \otimes_{\mathbb{k}} F$ with respect to the norm induced by the inner product $\langle m \otimes n, m' \otimes n' \rangle = \langle m, m' \rangle \langle n, n' \rangle$.

Given two inner product spaces $E$ and $F$ over a field $\mathbb{k}$, their tensor product $E \otimes_{\mathbb{k}} F$ is nontrivial (i.e., it contains at least two distinct elements).

Given two inner product spaces $E$ and $F$ over a field $\mathbb{k}$ (where $\mathbb{k}$ is $\mathbb{R}$ or $\mathbb{C}$), their complete tensor product $E \otimes_{h, \mathbb{k}} F$ is nontrivial, meaning it contains at least two distinct elements.

Given two inner product spaces $E$ and $F$ over a field $\mathbb{k}$ (where $\mathbb{k}$ is $\mathbb{R}$ or $\mathbb{C}$), this function is the canonical embedding of the algebraic tensor product $E \otimes_{\mathbb{k}} F$ into its Cauchy completion, the complete (or Hilbert) tensor product $E \otimes_h F$.

Given two inner product spaces $E$ and $F$ over a field $\mathbb{k}$ (where $\mathbb{k}$ is $\mathbb{R}$ or $\mathbb{C}$), this instance defines the canonical inclusion (coercion) that allows an element of the algebraic tensor product $E \otimes_{\mathbb{k}} F$ to be automatically treated as an element of its Cauchy completion, the Hilbert tensor product $E \otimes_h F$.

Let $E$ and $F$ be inner product spaces over a field $\mathbb{k}$ (where $\mathbb{k}$ is $\mathbb{R}$ or $\mathbb{C}$). The image of the canonical embedding from the algebraic tensor product $E \otimes_{\mathbb{k}} F$ into the complete tensor product $E \otimes_h F$ is dense in $E \otimes_h F$.

Let $E$ and $F$ be inner product spaces over a field $\mathbb{k}$ (where $\mathbb{k}$ is $\mathbb{R}$ or $\mathbb{C}$). The image of the zero element $0$ of the algebraic tensor product $E \otimes_{\mathbb{k}} F$ under the canonical embedding into the complete tensor product $E \otimes_h F$ is the zero element $0$ of $E \otimes_h F$.

Let $\mathbb{k}$ be a field (either $\mathbb{R}$ or $\mathbb{C}$), and let $E$ and $F$ be inner product spaces over $\mathbb{k}$. For any element $x \in E \otimes_{\mathbb{k}} F$ in the algebraic tensor product, its image under the canonical embedding into the complete tensor product $E \otimes_{h, \mathbb{k}} F$ is zero if and only if $x = 0$.

Let $E$ and $F$ be inner product spaces over a field $\mathbb{k}$ (where $\mathbb{k}$ is $\mathbb{R}$ or $\mathbb{C}$). For any element $x$ in the algebraic tensor product $E \otimes_{\mathbb{k}} F$, the image of its negation under the canonical embedding into the complete tensor product $E \otimes_h F$ is equal to the negation of its image. That is, $(-x : E \otimes_h F) = -(x : E \otimes_h F)$.

For any elements $x$ and $y$ in the algebraic tensor product $E \otimes_{\mathbb{k}} F$, the image of their difference $x - y$ under the canonical embedding into the complete tensor product $E \otimes_h F$ is equal to the difference of their images in $E \otimes_h F$.

Let $E$ and $F$ be inner product spaces over a field $\mathbb{k}$ (where $\mathbb{k}$ is $\mathbb{R}$ or $\mathbb{C}$). For any elements $x$ and $y$ in the algebraic tensor product $E \otimes_{\mathbb{k}} F$, the canonical embedding into the complete tensor product $E \otimes_h F$ preserves addition, such that the image of their sum is equal to the sum of their images: $(x + y)_{E \otimes_h F} = x_{E \otimes_h F} + y_{E \otimes_h F}$.

Let $\mathbb{k}$ be a field (either $\mathbb{R}$ or $\mathbb{C}$), and let $E$ and $F$ be inner product spaces over $\mathbb{k}$. For any scalar $c \in \mathbb{k}$ and any element $x$ in the algebraic tensor product $E \otimes_{\mathbb{k}} F$, the canonical embedding into the complete tensor product $E \otimes_h F$ preserves scalar multiplication, such that the image of the scalar product is equal to the scalar product of the image: $$(c \cdot x)_{E \otimes_h F} = c \cdot x_{E \otimes_h F}$$

Let $E$ and $F$ be inner product spaces over a field $\mathbb{k}$ (where $\mathbb{k}$ is $\mathbb{R}$ or $\mathbb{C}$). For any elements $x$ and $y$ in the algebraic tensor product $E \otimes_{\mathbb{k}} F$, the inner product of their images under the canonical embedding into the complete tensor product $E \otimes_h F$ is equal to their original inner product in $E \otimes_{\mathbb{k}} F$: $$\langle x, y \rangle_{E \otimes_h F} = \langle x, y \rangle_{E \otimes_{\mathbb{k}} F}$$

Let $E$ and $F$ be inner product spaces over a field $\mathbb{k}$ (where $\mathbb{k}$ is $\mathbb{R}$ or $\mathbb{C}$). For any element $x$ in the algebraic tensor product $E \otimes_{\mathbb{k}} F$, the norm of its image under the canonical embedding into the complete tensor product $E \otimes_h F$ is equal to its original norm in $E \otimes_{\mathbb{k}} F$: $$\|x\|_{E \otimes_h F} = \|x\|_{E \otimes_{\mathbb{k}} F}$$

Let $E$ and $F$ be inner product spaces over a field $\mathbb{k}$ (where $\mathbb{k}$ is either $\mathbb{R}$ or $\mathbb{C}$). The function is the canonical linear isometric embedding from the algebraic tensor product $E \otimes_{\mathbb{k}} F$ into its completion, the complete tensor product $E \otimes_h F$.

Let $E$ and $F$ be inner product spaces over a field $\mathbb{k}$ (where $\mathbb{k}$ is $\mathbb{R}$ or $\mathbb{C}$). The function underlying the canonical linear isometric embedding $tIncl_{\ell i} : E \otimes_{\mathbb{k}} F \to E \otimes_h F$ is identical to the canonical embedding (coercion) $coe' : E \otimes_{\mathbb{k}} F \to E \otimes_h F$ from the algebraic tensor product to the complete tensor product.

Let $E$ and $F$ be inner product spaces over a field $\mathbb{k}$ (where $\mathbb{k}$ is either $\mathbb{R}$ or $\mathbb{C}$). This function is the canonical embedding of the algebraic tensor product $E \otimes_{\mathbb{k}} F$ into its completion, the complete tensor product $E \otimes_h F$, regarded as a continuous linear map.

Let $E$ and $F$ be inner product spaces over a field $\mathbb{k}$ (where $\mathbb{k}$ is either $\mathbb{R}$ or $\mathbb{C}$). The function underlying the canonical continuous linear embedding $tInclL : E \otimes_{\mathbb{k}} F \to E \otimes_h F$ is identical to the canonical embedding (coercion) $coe' : E \otimes_{\mathbb{k}} F \to E \otimes_h F$ from the algebraic tensor product to the complete tensor product $E \otimes_h F$.

Let $\mathbb{k}$ be a field (either $\mathbb{R}$ or $\mathbb{C}$) and let $E$ and $F$ be nontrivial inner product spaces over $\mathbb{k}$. Let $\iota : E \otimes_{\mathbb{k}} F \to E \otimes_h F$ be the canonical continuous linear embedding of the algebraic tensor product into its completion, the complete tensor product. Then the operator norm of this embedding is equal to $1$, i.e., $\|\iota\| = 1$.

Let $\mathbb{k}$ be a field (either $\mathbb{R}$ or $\mathbb{C}$), and let $E$ and $F$ be inner product spaces over $\mathbb{k}$. Let $E \otimes_h F$ denote the complete (Hilbert) tensor product of $E$ and $F$. Suppose $P$ is a property (predicate) on $E \otimes_h F$ such that: 1. $P(0)$ holds. 2. For every $x \in E$ and $y \in F$, the property holds for the pure tensor, $P(x \otimes y)$. 3. For any algebraic tensors $u, v \in E \otimes F$, if $P(u)$ and $P(v)$ hold, then $P(u + v)$ holds for their sum in the completion. 4. The set of elements $\{z \in E \otimes_h F \mid P(z)\}$ is a closed set in the topology of $E \otimes_h F$. Then, $P(z)$ holds for all $z \in E \otimes_h F$.

Let $E$ and $F$ be inner product spaces over a field $\mathbb{k}$ (where $\mathbb{k}$ is $\mathbb{R}$ or $\mathbb{C}$). There exists a linear isometric equivalence between their complete tensor products $E \otimes_h F$ and $F \otimes_h E$. This equivalence is the unique continuous extension of the algebraic commutativity map $x \otimes y \mapsto y \otimes x$.

Let $\mathbb{k}$ be a field (either $\mathbb{R}$ or $\mathbb{C}$), and let $E$ and $F$ be inner product spaces over $\mathbb{k}$. Let $\text{comm}_{E, F} : E \otimes_h F \simeq F \otimes_h E$ be the linear isometric equivalence defining the commutativity of the complete tensor product. The inverse of this equivalence, $(\text{comm}_{E, F})^{-1}$, is equal to the commutativity equivalence for the swapped spaces, $\text{comm}_{F, E} : F \otimes_h E \simeq E \otimes_h F$.

Let $E$ and $F$ be inner product spaces over a field $\mathbb{k}$ (where $\mathbb{k}$ is $\mathbb{R}$ or $\mathbb{C}$). Let $\text{comm}_h : E \otimes_h F \simeq F \otimes_h E$ be the linear isometric equivalence between their complete tensor products, and let $\text{comm}_{alg} : E \otimes_{\mathbb{k}} F \simeq F \otimes_{\mathbb{k}} E$ be the commutativity map for their algebraic tensor products. For any element $x$ in the algebraic tensor product $E \otimes_{\mathbb{k}} F$, it holds that $$\text{comm}_h(x) = \text{comm}_{alg}(x),$$ where the $x$ on the left-hand side is understood as its image under the canonical embedding into the complete tensor product.

Let $R$ be a commutative semiring and $\mathbb{k}$ be a field that is either the real numbers $\mathbb{R}$ or complex numbers $\mathbb{C}$ (an `RCLike` field). Let $\sigma : R \to \mathbb{k}$ be a surjective ring homomorphism. Suppose $M$ and $N$ are modules over $R$, and $E$ and $F$ are inner product spaces over $\mathbb{k}$. Let $f : M \to E$ and $g : N \to F$ be $\sigma$-semilinear maps. If $f$ and $g$ both have dense ranges in their respective codomains, then the induced tensor product map $f \otimes g : M \otimes_R N \to E \otimes_{\mathbb{k}} F$ also has a dense range.

Let $\mathbb{k}$ be a field (either $\mathbb{R}$ or $\mathbb{C}$) and let $E$, $F$, and $G$ be inner product spaces over $\mathbb{k}$. The complete tensor product $\otimes_h$ is associative, meaning there exists a canonical linear isometric equivalence between $(E \otimes_h F) \otimes_h G$ and $E \otimes_h (F \otimes_h G)$.