LemmaStatement: AI-generatedProof: AI-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Realification of Hilbert pairings and conjugate-linear coefficients

Statement

Assume the Axiom of Countable Choice. Let $H$ be a complex Hilbert space with inner product linear in the first variable. Its realification $H_{R}$ is the same additive group with scalar multiplication restricted to real scalars. With $(u, v)_{R} = Re ⟨ u, v ⟩_{H}$ , it is a real Hilbert space with the same norm and metric as $H$ . Every bounded real-linear functional $f : H_{R} \to R$ has a unique representative $y \in H$ such that $f (u) = (u, y)_{R}$ for all $u$ , and $∥ f ∥ = ∥ y ∥$ . Here $∥ f ∥ : = \sup {∣ f (u) ∣ : ∥ u ∥ \leq 1}$ is the norm in the real normed dual of $H_{R}$ .

In particular, let $(X, A, μ)$ be a measure space and $H = L^{2} (μ; C)$ , the space of square-integrable measurable complex functions modulo equality almost everywhere, with $⟨ z, v ⟩ = \int_{X} z \overline{v} d μ$ . Let $a, b$ be measurable essentially bounded complex functions, with $a$ real almost everywhere. Write $∥ c ∥_{\infty} : = ∥ ∣ c ∣ ∥_{\infty, μ}$ for $c \in {a, b}$ , where the right-hand side is the essential supremum norm of the real-valued measurable function $∣ c ∣$ . Define the bounded real-linear operator $L$ by $L z = i (a z + b \overline{z})$ . Its real Hilbert adjoint, defined by $(L z, v)_{R} = (z, L_{R}^{*} v)_{R}$ for every $z, v$ , is $L_{R}^{*} v = - i a v + i b \overline{v}$ . Consequently $(L z, v)_{R} + (z, - L_{R}^{*} v)_{R} = 0$ for every $z, v \in H$ .

Statement references

Realification of a complex vector space by restriction of scalars

Real and complex inner product spaces, with the inner product linear in the first argument

The norm ‖ v‖=√⟨ v,v⟩ induced by a real or complex inner product

Hilbert space

The Axiom of Countable Choice (AC_ω)

L² with the integral pairing is a Hilbert space

The Hilbert-space adjoint of a bounded operator

The dual space X^* of a normed space and its dual norm

A bounded linear operator between normed spaces

Measure spaces

The essential supremum of a measurable function with respect to a measure

Complex Lp classes and Euclidean test-function conventions

Facts & Assumptions

[F1]

Realification preserves the underlying set and addition and restricts scalar multiplication to real scalars; it preserves the original norm and induced metric (Realification of a complex vector space by restriction of scalars, Realification preserves a complex norm and its metric).

[F2]

The complex inner product is linear in its first variable, conjugate symmetric and positive definite, and its norm is the square root of its diagonal value. A Hilbert space is complete for the metric induced by this norm (Real and complex inner product spaces, with the inner product linear in the first argument, The norm ‖ v‖=√⟨ v,v⟩ induced by a real or complex inner product, Hilbert space).

[F3]

Under Countable Choice, every bounded linear functional on a real Hilbert space has a unique representing vector with equal norm (The Axiom of Countable Choice (AC_ω), Riesz representation for Hilbert spaces).

[F4]

Under Countable Choice, complex $L^{2} (μ)$ is a Hilbert space with the stated integral pairing; products of two $L^{2}$ functions are integrable and the pairing is independent of representatives (L² with the integral pairing is a Hilbert space).

[F5]

For a real-valued measurable function $r$ with finite essential supremum norm, $∣ r ∣ \leq ∥ r ∥_{\infty, μ}$ almost everywhere (The essential supremum of a measurable function with respect to a measure, The essential supremum is attained as the least essential bound). Applying this to $r = ∣ a ∣$ and $r = ∣ b ∣$ gives $∣ a ∣ \leq ∥ a ∥_{\infty}$ and $∣ b ∣ \leq ∥ b ∥_{\infty}$ almost everywhere, with the coefficient norms specified in the Statement.

[F6]

The real Hilbert adjoint of a bounded real-linear operator is uniquely characterized by its pairing identity (The Hilbert-space adjoint of a bounded operator).

[F7]

Bounded real-linear functionals on $H_{R}$ form its real normed dual, with norm $∥ f ∥ = \sup_{∥ u ∥ \leq 1} ∣ f (u) ∣$ (The dual space X^* of a normed space and its dual norm). A real-linear operator $T$ is bounded when $∥ T u ∥ \leq C ∥ u ∥$ for all $u$ and some real $C \geq 0$ (A bounded linear operator between normed spaces).

[F8]

On a measure space, complex measurability is componentwise; moduli, conjugates and products of measurable complex functions are measurable. Complex $L^{2}$ consists of square-integrable such functions modulo equality almost everywhere, with norm determined by the integral of the squared modulus (Complex Lp classes and Euclidean test-function conventions).

[F9]

On an arbitrary measure space the nonnegative integral is monotone, is homogeneous for positive finite scalars and has value zero on the zero function, and is additive on nonnegative measurable functions (Monotonicity and nonnegative homogeneity of the nonnegative integral, Additivity of the nonnegative Lebesgue integral).

[F10]

A nonnegative measurable function has integral zero if and only if it vanishes almost everywhere, without any finiteness assumption on its integral beforehand (A nonnegative measurable function has integral 0 exactly when it vanishes almost everywhere).

Proof

[F1] Realification preserves the underlying set and addition and restricts scalar multiplication to real scalars; it preserves the original norm and induced metric (Realification of a complex vector space by restriction of scalars, Realification preserves a complex norm and its metric).

[F2] The complex inner product is linear in its first variable, conjugate symmetric and positive definite, and its norm is the square root of its diagonal value. A Hilbert space is complete for the metric induced by this norm (Real and complex inner product spaces, with the inner product linear in the first argument, The norm ‖ v‖=√⟨ v,v⟩ induced by a real or complex inner product, Hilbert space).

[F3] Under Countable Choice, every bounded linear functional on a real Hilbert space has a unique representing vector with equal norm (The Axiom of Countable Choice (AC_ω), Riesz representation for Hilbert spaces).

[F4] Under Countable Choice, complex $L^{2} (μ)$ is a Hilbert space with the stated integral pairing; products of two $L^{2}$ functions are integrable and the pairing is independent of representatives (L² with the integral pairing is a Hilbert space).

[F5] For a real-valued measurable function $r$ with finite essential supremum norm, $∣ r ∣ \leq ∥ r ∥_{\infty, μ}$ almost everywhere (The essential supremum of a measurable function with respect to a measure, The essential supremum is attained as the least essential bound). Applying this to $r = ∣ a ∣$ and $r = ∣ b ∣$ gives $∣ a ∣ \leq ∥ a ∥_{\infty}$ and $∣ b ∣ \leq ∥ b ∥_{\infty}$ almost everywhere, with the coefficient norms specified in the Statement.

[F6] The real Hilbert adjoint of a bounded real-linear operator is uniquely characterized by its pairing identity (The Hilbert-space adjoint of a bounded operator).

[F7] Bounded real-linear functionals on $H_{R}$ form its real normed dual, with norm $∥ f ∥ = \sup_{∥ u ∥ \leq 1} ∣ f (u) ∣$ (The dual space X^* of a normed space and its dual norm). A real-linear operator $T$ is bounded when $∥ T u ∥ \leq C ∥ u ∥$ for all $u$ and some real $C \geq 0$ (A bounded linear operator between normed spaces).

[F8] On a measure space, complex measurability is componentwise; moduli, conjugates and products of measurable complex functions are measurable. Complex $L^{2}$ consists of square-integrable such functions modulo equality almost everywhere, with norm determined by the integral of the squared modulus (Complex Lp classes and Euclidean test-function conventions).

[F9] On an arbitrary measure space the nonnegative integral is monotone, is homogeneous for positive finite scalars and has value zero on the zero function, and is additive on nonnegative measurable functions (Monotonicity and nonnegative homogeneity of the nonnegative integral, Additivity of the nonnegative Lebesgue integral).

[F10] A nonnegative measurable function has integral zero if and only if it vanishes almost everywhere, without any finiteness assumption on its integral beforehand (A nonnegative measurable function has integral 0 exactly when it vanishes almost everywhere).

Given: Countable Choice, a complex Hilbert space $H$ with first-variable-linear inner product, and, for the coefficient assertion, the measure space and functions $a, b$ in the statement.

Goal: Establish the real Hilbert and representation assertions and compute the real adjoint on $L^{2} (μ; C)$ .

1.1F1F2

Restriction of scalars gives a real vector space. Taking real parts of the complex inner-product identities gives real linearity in the first variable and symmetry; symmetry then gives linearity in the second. Moreover $(u, u)_{R} = ⟨ u, u ⟩_{H} \geq 0$ , with equality only for $u = 0$ . Thus this is a real inner product and its induced norm is the original norm.

1.2F4F5F7F8F9F10algebra

On complex $L^{2}$ , conjugation preserves measurability and the $L^{2}$ norm. For $c \in {a, b}$ , put $M = ∥ c ∥_{\infty}$ and let $u \in L^{2}$ have a finite-valued measurable representative. The function $q = \max {∣ c u ∣^{2} - M^{2} ∣ u ∣^{2}, 0}$ is nonnegative measurable and vanishes almost everywhere by the essential bound. Thus $\int q d μ = 0$ by F10, before assuming integrability of $∣ c u ∣^{2}$ . The pointwise inequality $∣ c u ∣^{2} \leq M^{2} ∣ u ∣^{2} + q$ , followed by monotonicity, additivity and homogeneity, gives $\int ∣ c u ∣^{2} d μ \leq M^{2} \int ∣ u ∣^{2} d μ < \infty$ ; when $M = 0$ , the zero-function clause gives the same inequality, and $\int ∣ u ∣^{2}$ is finite. Hence $∥ c u ∥_{2} \leq M ∥ u ∥_{2}$ . Taking $u = z, \overline{z}, v, \overline{v}$ as needed shows that $a z$ , $b \overline{z}$ , $a v$ and $b \overline{v}$ belong to $L^{2}$ . The triangle inequality now gives $∥ L z ∥_{2} \leq (∥ a ∥_{\infty} + ∥ b ∥_{\infty}) ∥ z ∥_{2}$ , and the same bound holds for the proposed adjoint applied to $v$ . These operations respect equality almost everywhere; $L$ and the proposed adjoint are bounded and real-linear.

2.1F1F2F3F7step 1.1

The induced metric is therefore the original metric, so every realified Cauchy sequence converges because $H$ is complete. The realification is a real Hilbert space. Applying real Riesz representation gives the unique vector $y$ and the norm equality for every bounded real-linear $f$ .

2.2F4step 1.2algebra

For arbitrary $z, v \in L^{2}$ , the integrands below are integrable. Since $a = \overline{a}$ almost everywhere and a complex number and its conjugate have the same real part, $Re \int i a z \overline{v} = Re \int z \overline{(- i a v)}$ and $Re \int i b \overline{z} \overline{v} = Re \int z \overline{(i b \overline{v})}$ . Adding the identities gives $(L z, v)_{R} = (z, - i a v + i b \overline{v})_{R}$ .

3.1F6step 1.2step 2.2algebra∎

By the defining uniqueness of the real adjoint, the vector on the right in step 2.2 is $L_{R}^{*} v$ . Moving that pairing to the other side gives the cancellation identity.

Remarks

The coefficient calculation is an identity for bounded operators on $L^{2}$ ; it makes no assertion about the domain of a Laplacian, evolution, or extensions of pairings to other spaces.

Depends on

Used by · 4

Dependency tree · 2 levels

Each result is drawn at its shortest distance from this item. Arrows run from prerequisites to the items that use them; this item has a heavier outline. External source nodes link directly to the cited text. The view shows up to two dependency steps; open a prerequisite to continue.