PhyslibAlpha.ProbabilisticTheory.HilbertSpace.Unbounded.UnitaryInfra.SesquilinearForm

For a compact positive linear functional $\Lambda$ defined on the space of compactly supported continuous functions $C_c(X, \mathbb{R})$, this is the unique regular Borel measure $\mu$ on $X$ such that for every $f \in C_c(X, \mathbb{R})$, the integral of $f$ with respect to $\mu$ equals the value of the functional applied to $f$: $$\int_X f \, d\mu = \Lambda(f)$$

Let $X$ be a topological space and $\Lambda$ be a compact positive linear functional on the space of compactly supported continuous functions $C_c(X, \mathbb{R})$. Let $\mu$ be the unique Borel measure on $X$ such that for every $f \in C_c(X, \mathbb{R})$, the integral of $f$ with respect to $\mu$ is equal to the value of the functional applied to $f$: $$\int_X f \, d\mu = \Lambda(f)$$ Then, the measure $\mu$ is a regular measure.

Let $X$ be a topological space and $\Lambda$ be a compact positive linear functional on the space of compactly supported continuous functions $C_c(X, \mathbb{R})$. Let $\mu$ be the regular Borel measure associated with $\Lambda$. For any function $f \in C_c(X, \mathbb{R})$, the integral of $f$ with respect to $\mu$ is equal to the value of the functional $\Lambda$ applied to $f$: $$\int_X f \, d\mu = \Lambda(f)$$

Let $X$ be a topological space and $\Lambda$ be a compact positive functional on the space of compactly supported continuous functions $C_c(X, \mathbb{R})$. Let $\mu$ be the regular Borel measure associated with $\Lambda$, defined such that $\int_X g \, d\mu = \Lambda(g)$ for all $g \in C_c(X, \mathbb{R})$. For any function $f \in C_c(X, \mathbb{R})$ and any measurable set $S \subseteq X$, if $f(x) = 0$ for all $x \in S$, then the integral of $f$ with respect to the restriction of the measure $\mu$ to $S$ is zero: $$\int_X f \, d(\mu|_S) = 0$$

Let $X$ be a topological space. Let $\Lambda_1$ and $\Lambda_2$ be compact positive linear functionals on the space of compactly supported continuous functions $C_c(X, \mathbb{R})$. If $\Lambda_1(f) = \Lambda_2(f)$ for all $f \in C_c(X, \mathbb{R})$, then their associated regular Borel measures $\mu_1$ and $\mu_2$ (defined such that $\int_X f \, d\mu_i = \Lambda_i(f)$) are equal.

Given a continuous linear map $L: F \to G$ and a continuous bilinear map $B: E \to \mathcal{L}(G, K)$ (where $\mathcal{L}(G, K)$ is the space of continuous linear maps from $G$ to $K$ over $\mathbb{R}$), this definition constructs a new continuous bilinear map $E \to \mathcal{L}(F, K)$. For any $e \in E$, the resulting linear map from $F$ to $K$ is the composition of $L$ followed by the linear map $B(e)$. In other words, it maps $(e, f)$ to $B(e, L(f))$.

Let $X$ be a measurable space, and let $E, F, G,$ and $K$ be real normed spaces. Let $\mu$ be a vector measure on $X$ with values in $F$, $L: F \to G$ be a continuous $\mathbb{R}$-linear map, and $B: E \to \mathcal{L}_{\mathbb{R}}(G, K)$ be a continuous bilinear pairing. Let $L \circ \mu$ denote the vector measure taking values in $G$ obtained by composing $\mu$ with $L$ (i.e., $(L \circ \mu)(A) = L(\mu(A))$ for any measurable set $A$). Then, the transpose of $L \circ \mu$ with respect to the pairing $B$ is equal to the transpose of $\mu$ with respect to the pullback pairing $B \circ L$, where the pullback pairing is defined as $(e, f) \mapsto B(e, L(f))$.

Let $X$ be a measurable space and $E, F, G, K$ be normed real vector spaces. Let $\mu$ be an $F$-valued vector measure on $X$, $L: F \to G$ be a continuous $\mathbb{R}$-linear map, and $B: E \to \mathcal{L}(G, K)$ be a continuous $\mathbb{R}$-bilinear map. Let $f: X \to E$ be a function. If $f$ is integrable with respect to $\mu$ and also integrable with respect to the transformed measure $L \circ \mu$ (the vector measure obtained by post-composing $\mu$ with $L$), then: $$\int_X f \, d[B; L \circ \mu] = \int_X f \, d[B_L; \mu]$$ where $B_L: E \to \mathcal{L}(F, K)$ is the "pulled-back" bilinear pairing defined by $B_L(e, v) = B(e, L(v))$.

This is the continuous $\mathbb{R}$-bilinear pairing between complex numbers and real numbers. Specifically, it is the map $\mathbb{C} \to \mathcal{L}_{\mathbb{R}}(\mathbb{R}, \mathbb{C})$ which, for a complex number $z \in \mathbb{C}$ and a real number $r \in \mathbb{R}$, returns the product $z \cdot r \in \mathbb{C}$. This pairing is used to integrate complex-valued functions against real signed measures.

Let $E, F, G$, and $K$ be normed real vector spaces. Given a continuous bilinear pairing $B : E \to \mathcal{L}(F, G)$ and a continuous linear map $C : G \to K$, the function returns a new continuous bilinear pairing $B' : E \to \mathcal{L}(F, K)$ defined by $B'(e)(f) = C(B(e)(f))$ for all $e \in E$ and $f \in F$.

Let $X$ be a measurable space, and let $E, F, G$, and $K$ be normed real vector spaces. Let $\mu$ be an $F$-valued vector measure on $X$. Given a continuous bilinear pairing $B: E \to \mathcal{L}(F, G)$ and a continuous linear map $C: G \to K$, we define a post-composed pairing $B': E \to \mathcal{L}(F, K)$ by $B'(x)(f) = C(B(x)(f))$. For any set $s \subseteq X$ and vector $x \in E$, the transpose operation satisfies the following identity: $$(\mu.\text{transpose } B') s x = C((\mu.\text{transpose } B) s x)$$ In this context, the transpose operation $(\mu.\text{transpose } B) s x$ represents the evaluation of the pairing $B$ between the vector $x$ and the value of the measure $\mu$ on the set $s$.

Let $X$ be a measurable space and $E, F$, and $G$ be normed real vector spaces. Let $T : \text{Set } X \to \mathcal{L}_{\mathbb{R}}(E, F)$ and $T' : \text{Set } X \to \mathcal{L}_{\mathbb{R}}(E, G)$ be set functions taking values in the space of continuous linear maps. Let $C : F \to L_{\mathbb{R}} G$ be a continuous linear map. If for every measurable set $s \subseteq X$ and every vector $x \in E$, the condition $T'(s)x = C(T(s)x)$ is satisfied, then for any $L^1$ simple function $f : X \to E$ with respect to a measure $\mu$, we have: $$\text{setToL1S}(T', f) = C(\text{setToL1S}(T, f))$$ where $\text{setToL1S}$ is the operator that integrates a simple function with respect to a set-indexed linear operator.

Let $X$ be a measurable space with a measure $\mu$. Let $E$ be a real normed space, and let $F$ and $G$ be real Banach spaces. Suppose $T: \text{Set } X \to \mathcal{L}(E, F)$ and $T': \text{Set } X \to \mathcal{L}(E, G)$ are dominated finite-measure additive set functions (with respect to $\mu$) that map sets to continuous linear operators. If $C: F \to G$ is a continuous linear map such that for every measurable set $s$ and every $x \in E$, the condition $T'(s)(x) = C(T(s)(x))$ holds, then for any function $f$ in the $L^1$ space $L^1(X, \mu; E)$, the $L^1$ integration operator `setToL1` satisfies: $$\text{setToL1}(T', f) = C(\text{setToL1}(T, f))$$

Let $X$ be a measurable space equipped with a measure $\mu$. Let $E$, $F$, and $G$ be real normed spaces, where $F$ and $G$ are complete. Suppose $T: \mathcal{P}(X) \to \mathcal{L}(E, F)$ and $T': \mathcal{P}(X) \to \mathcal{L}(E, G)$ are additive set functions that are dominated by $\mu$ (specifically, they satisfy the `DominatedFinMeasAdditive` condition). Let $C: F \to G$ be a continuous linear map such that for every measurable set $s \subseteq X$ and every $x \in E$, $T'(s)x = C(T(s)x)$. Then for any $\mu$-integrable function $f: X \to E$, the generalized integral of $f$ with respect to $T'$ satisfies $$\int_X f \, d T' = C \left( \int_X f \, d T \right)$$ where $\int_X f \, d T$ denotes the value of `MeasureTheory.setToFun` for the function $f$ and set function $T$.

Let $X$ be a measurable space. Let $E, F, G$, and $K$ be normed real vector spaces, where $G$ and $K$ are complete (Banach spaces). Let $\mu$ be an $F$-valued vector measure on $X$. Let $B : E \to \mathcal{L}(F, G)$ be a continuous bilinear pairing and $C : G \to K$ be a continuous linear map. If $f : X \to E$ is a function integrable with respect to $\mu$, then applying $C$ to the vector integral of $f$ with respect to $\mu$ via the pairing $B$ is equal to the vector integral of $f$ with respect to $\mu$ via the postcomposed pairing $C \circ B$: $$C \left( \int_X f(x) \, d[B; \mu] \right) = \int_X f(x) \, d[C \circ B; \mu]$$ where the pairing $C \circ B$ is defined by $(C \circ B)(e)(v) = C(B(e)(v))$ for $e \in E$ and $v \in F$.

The continuous $\mathbb{R}$-linear map from the real numbers $\mathbb{R}$ to the complex numbers $\mathbb{C}$ that maps each real number $r$ to the purely imaginary number $ir$, where $i$ is the imaginary unit.

For any real number $r \in \mathbb{R}$, the continuous $\mathbb{R}$-linear map `imaginaryOfRealCLM` applied to $r$ yields $ir$, where $i$ is the imaginary unit.

Let $Y$ be a measurable space. For any two signed measures $s$ and $t$ on $Y$, the complex measure constructed from $s$ and $t$ (denoted as $s + it$) is equal to the sum of the signed measure $s$ mapped into the complex numbers via the natural inclusion $r \mapsto r$ and the signed measure $t$ mapped into the complex numbers via the map $r \mapsto ir$.

Let $(X, \mathcal{A})$ be a measurable space and let $s$ and $t$ be real-valued signed measures on $X$. Let $\nu = s + it$ be the complex measure constructed such that $s$ is its real part and $t$ is its imaginary part. For any complex-valued function $f: X \to \mathbb{C}$, if $f$ is integrable with respect to the measures $s$ and $it$ (viewed as complex measures via the embeddings $r \mapsto r$ and $r \mapsto ir$ respectively), then the integral of $f$ with respect to $\nu$ is the sum of the integrals of $f$ with respect to these components: $$\int_X f(x) \, d(s + it)(x) = \int_X f(x) \, ds(x) + \int_X f(x) \, d(it)(x)$$ In the formal statement, the measure $s$ is embedded into the complex plane via the map $r \mapsto r + 0i$ (`ofRealCLM`) and $t$ is embedded via the map $r \mapsto 0 + ri$ (`imaginaryOfRealCLM`).

Let $\iota : \mathbb{R} \to \mathbb{C}$ be the standard inclusion of real numbers into the complex field. Let $B : \mathbb{C} \times \mathbb{C} \to \mathbb{C}$ be the continuous $\mathbb{R}$-bilinear map representing complex multiplication, defined by $(z, w) \mapsto z \cdot w$. The pullback of $B$ along $\iota$ in its second argument—which is the map $(z, r) \mapsto B(z, \iota(r))$—is equal to the standard continuous $\mathbb{R}$-bilinear pairing $\mathbb{C} \times \mathbb{R} \to \mathbb{C}$ given by $(z, r) \mapsto z \cdot r$.

Let $(X, \mathcal{A})$ be a measurable space and $\mu$ be a finite measure on $X$. For any complex-valued function $f: X \to \mathbb{C}$ that is integrable with respect to $\mu$, the integral of $f$ with respect to the complex measure obtained by mapping the real-valued signed measure $\mu$ into the complex plane via the natural embedding $\mathbb{R} \hookrightarrow \mathbb{C}$ is equal to the integral of $f$ with respect to $\mu$ itself. That is, $$\int_X f(x) \, d(\iota \circ \mu)(x) = \int_X f(x) \, d\mu(x)$$ where $\iota: \mathbb{R} \to \mathbb{C}$ is the standard inclusion $r \mapsto r + 0i$.

Let $(X, \mathcal{M})$ be a measurable space and $\mu$ be a finite measure on $X$. Let $L: \mathbb{R} \to \mathbb{C}$ be a continuous $\mathbb{R}$-linear map that is an isometry, such that $\|L(r)\| = |r|$ for all $r \in \mathbb{R}$. If a complex-valued function $f: X \to \mathbb{C}$ is integrable with respect to $\mu$, then $f$ is also integrable with respect to the complex measure obtained by mapping the values of $\mu$ (viewed as a signed measure) through $L$.

Let $X$ be a measurable space and $\mu$ be a finite measure on $X$. Let $f : X \to \mathbb{C}$ be a complex-valued function. If $f$ is integrable with respect to $\mu$, then $f$ is also integrable with respect to the complex-valued measure obtained by post-composing the real-valued signed measure $\mu$ with the natural inclusion map $\iota : \mathbb{R} \to \mathbb{C}$ (where $\iota(r) = r + 0i$).

Let $X$ be a measurable space and $\mu$ be a finite measure on $X$. Let $f : X \to \mathbb{C}$ be a complex-valued function. If $f$ is integrable with respect to $\mu$, then $f$ is also integrable with respect to the complex-valued measure obtained by post-composing the real-valued signed measure $\mu$ with the continuous $\mathbb{R}$-linear map $L: \mathbb{R} \to \mathbb{C}$ defined by $L(r) = ir$.

Let $X$ be a measurable space and let $\mu$ and $\nu$ be finite measures on $X$. Let $L: \mathbb{R} \to \mathbb{C}$ be a continuous real-linear map. The complex-valued measure obtained by post-composing the signed measure $\mu - \nu$ with $L$ is equal to the difference of the complex-valued measures obtained by post-composing $\mu$ and $\nu$ with $L$ individually: $$L \circ (\mu - \nu) = (L \circ \mu) - (L \circ \nu)$$ where $L \circ \rho$ denotes the measure defined by $(L \circ \rho)(A) = L(\rho(A))$ for any measurable set $A \subseteq X$.

Let $(X, \mathcal{M})$ be a measurable space and let $\mu$ and $\nu$ be finite measures on $X$. Let $c \in \mathbb{R}$ be a real scalar and $f: X \to \mathbb{C}$ be a complex-valued function. If $f$ is integrable with respect to both $\mu$ and $\nu$, then the integral of $f$ with respect to the complex-valued measure obtained by scaling the signed difference $\mu - \nu$ by $c$ is equal to the scaled difference of the individual integrals: $$\int_X f \, d(c(\mu - \nu)) = c \left( \int_X f \, d\mu - \int_X f \, d\nu \right)$$ where the integral on the left is understood as an integral against a complex measure.

The function is the continuous $\mathbb{R}$-linear map from $\mathbb{C}$ to $\mathbb{C}$ defined by the assignment $z \mapsto i z$, where $i$ denotes the imaginary unit.

For any complex number $z \in \mathbb{C}$, the continuous $\mathbb{R}$-linear map $\text{imaginaryMulCLM}$ evaluated at $z$ is equal to $i z$, where $i$ denotes the imaginary unit.

The continuous $\mathbb{R}$-bilinear map from $\mathbb{C} \times \mathbb{R}$ to $\mathbb{C}$ defined by the assignment $(z, r) \mapsto i \cdot z \cdot r$, where $z$ is a complex number, $r$ is a real number, and $i$ denotes the imaginary unit. This map is obtained by post-composing the standard complex-real multiplication pairing with the linear map that performs multiplication by $i$.

The pullback of the continuous scalar multiplication map $B: \mathbb{C} \times \mathbb{C} \to \mathbb{C}$ (defined by $(z, w) \mapsto zw$) along the map $L: \mathbb{R} \to \mathbb{C}$ (defined by $r \mapsto ir$) in the second argument is equal to the continuous $\mathbb{R}$-bilinear pairing $\mathbb{C} \times \mathbb{R} \to \mathbb{C}$ defined by $(z, r) \mapsto izr$, where $i$ is the imaginary unit.

Let $X$ be a measurable space and $\mu$ be a finite measure on $X$. Let $f: X \to \mathbb{C}$ be a complex-valued function that is integrable with respect to $\mu$. If we define a complex-valued measure $i\mu$ by mapping the values of the signed measure $\mu$ through the purely imaginary map $r \mapsto ir$, then the integral of $f$ with respect to this measure satisfies: $$\int_X f \, d(i\mu) = i \int_X f \, d\mu$$ where $i$ is the imaginary unit.

Let $X$ be a measurable space and let $\mu$ and $\nu$ be finite measures on $X$. For any real scalar $c \in \mathbb{R}$ and any complex-valued function $f: X \to \mathbb{C}$ that is integrable with respect to both $\mu$ and $\nu$, the integral of $f$ with respect to the complex measure obtained by mapping the signed measure $c(\mu - \nu)$ through the map $r \mapsto ir$ is: $$\int_X f \, d(i \cdot c(\mu - \nu)) = c \cdot i \left( \int_X f \, d\mu - \int_X f \, d\nu \right)$$ where $i$ is the imaginary unit.

Let $X$ be a measurable space. If $s$ and $t$ are signed measures on $X$ such that their total variations are finite, then the complex measure formed by $s$ as the real part and $t$ as the imaginary part is also a finite complex measure (i.e., its total variation is finite).

Let $H$ be a complex Hilbert space. For any bounded linear operator $A$ on $H$ such that $0 \le A \le I$ in the Loewner order (where $I$ is the identity operator), and for any vector $x \in H$, the following inequality holds: $$\|Ax\|^2 \le \text{Re} \langle x, Ax \rangle$$ where $\langle \cdot, \cdot \rangle$ denotes the inner product on $H$.

Given a bounded normal operator $U$ on a complex Hilbert space $H$, let $\sigma(U)$ denote its spectrum. This function maps a real-valued, compactly supported continuous function $f: \sigma(U) \to \mathbb{R}$ to its corresponding complex-valued continuous function in $C(\sigma(U), \mathbb{C})$. This explicit mapping is used to facilitate the Riesz–Markov–Kakutani representation theorem by avoiding implicit coercions between real and complex scalars.

Let $U$ be a bounded normal operator on a complex Hilbert space, and let $\sigma(U)$ denote its spectrum. Let $f: \sigma(U) \to \mathbb{R}$ be a real-valued, compactly supported continuous function. If $\iota(f): \sigma(U) \to \mathbb{C}$ is the image of $f$ under the real-to-complex embedding map, then for any point $z \in \sigma(U)$, the value of the resulting complex function evaluated at $z$ is equal to the value of the original function $f$ at $z$, i.e., $(\iota(f))(z) = f(z)$.

Let $U$ be a bounded normal operator on a complex Hilbert space and let $\sigma(U)$ denote its spectrum. Let $\iota: C_c(\sigma(U), \mathbb{R}) \to C(\sigma(U), \mathbb{C})$ be the map that embeds a real-valued compactly supported continuous function into the space of complex-valued continuous functions. For any two such functions $f$ and $g$, it holds that $$\iota(f + g) = \iota(f) + \iota(g).$$

Let $U$ be a bounded normal operator on a complex Hilbert space and $\sigma(U) \subset \mathbb{C}$ denote its spectrum. Let $\iota: C_c(\sigma(U), \mathbb{R}) \to C(\sigma(U), \mathbb{C})$ be the embedding that maps a real-valued, compactly supported continuous function on the spectrum to its complex-valued counterpart. For any real scalar $r \in \mathbb{R}$ and any function $f \in C_c(\sigma(U), \mathbb{R})$, the map $\iota$ satisfies the homogeneity property: $$\iota(r \cdot f) = r \cdot \iota(f)$$ where the scalar $r$ on the right-hand side is treated as a complex number.

Given a bounded normal operator $U$ on a complex Hilbert space $H$ and its spectrum $\sigma(U)$, this definition constructs the operator $f(U)$ by applying the continuous functional calculus to a real-valued continuous function $f: \sigma(U) \to \mathbb{R}$. Although $f$ is real-valued, it is treated as a complex-valued function to utilize the standard complex continuous functional calculus, resulting in a continuous linear operator $f(U) \in \mathcal{L}(H)$.

Let $H$ be a complex Hilbert space and $U$ be a bounded normal operator on $H$ with spectrum $\sigma(U)$. For any real-valued continuous function $f: \sigma(U) \to \mathbb{R}$ (with compact support), the operator $f(U)$ obtained via the continuous functional calculus is self-adjoint.

Let $U$ be a bounded normal operator on a complex Hilbert space $H$, and let $\sigma(U)$ denote its spectrum. Suppose $f: \sigma(U) \to \mathbb{R}$ is a continuous function with compact support. If $f(z) \geq 0$ for all $z \in \sigma(U)$, then the operator $f(U)$, defined via the continuous functional calculus, is a positive operator, meaning that $\langle x, f(U)x \rangle \geq 0$ for all $x \in H$.

Given a bounded normal operator $U$ on a complex Hilbert space $H$ with spectrum $\sigma(U)$, and a vector $x \in H$, this definition constructs a positive linear functional on the space of continuous real-valued functions $C(\sigma(U), \mathbb{R})$. The functional maps a function $f$ to the real part of the inner product $\text{Re} \langle x, f(U) x \rangle$. This functional represents the vector state of the continuous functional calculus restricted to real-valued test functions and provides the necessary input for the Riesz–Markov–Kakutani representation theorem to produce a scalar measure.

Let $H$ be a complex Hilbert space and $U$ be a bounded normal operator on $H$ with spectrum $\sigma(U)$. For any vector $x \in H$ and any continuous real-valued function $f: \sigma(U) \to \mathbb{R}$ with compact support, the value of the associated scalar functional $\Lambda_{U,x}$ evaluated at $f$ is given by $$\Lambda_{U,x}(f) = \text{Re} \langle x, f(U) x \rangle,$$ where $f(U)$ is the continuous linear operator obtained by applying the continuous functional calculus to $f$, and $\langle \cdot, \cdot \rangle$ denotes the inner product on $H$.

Given a bounded normal operator $U$ on a complex Hilbert space $H$ and a vector $x \in H$, this is the regular Borel measure $\mu_x$ on the spectrum $\sigma(U)$ obtained via the Riesz–Markov–Kakutani representation theorem. The measure is associated with the positive linear functional $f \mapsto \text{Re} \langle x, f(U) x \rangle$ for real-valued continuous functions $f \in C(\sigma(U), \mathbb{R})$. It satisfies the property that for any such $f$, the integral $\int_{\sigma(U)} f \, d\mu_x$ equals the real part of the expectation value of the operator $f(U)$ in the state $x$.

Let $H$ be a complex Hilbert space and $U$ be a bounded normal operator on $H$. For any vector $x \in H$, the scalar spectral measure $\mu_x$ (defined on the spectrum $\sigma(U)$ via the Riesz–Markov–Kakutani representation theorem) is a finite measure.

Let $H$ be a complex Hilbert space and $U$ be a bounded normal operator on $H$. For any vector $x \in H$ and any real-valued continuous function $f$ on the spectrum $\sigma(U)$, the integral of $f$ with respect to the scalar spectral measure $\mu_x$ is equal to the real part of the inner product of $x$ and $f(U)x$: $$\int_{\sigma(U)} f(z) \, d\mu_x(z) = \text{Re} \langle x, f(U)x \rangle$$ where $f(U)$ is the operator obtained via the continuous functional calculus and $\langle \cdot, \cdot \rangle$ denotes the inner product on $H$.