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A Bochner-integral forward map with weakly closed graph

Example

Assume the ultrafilter lemma, dependent choice, and HB. Give $Ω = {0, 1}$ its full sigma-algebra and probability measure $μ (A) = ∣ A ∣ / 2$ . For each $s \in R$ , let $g_{s} : Ω \to R$ be the simple function $g_{s} (0) = s$ , $g_{s} (1) = s^{2}$ , and define $F (s) = \int_{Ω} g_{s} d μ = (s + s^{2}) / 2$ . With $X = Y = D = R$ and the usual real Hilbert structure, $D$ is weakly closed and $F$ is nonlinear; if $s_{n} ⇀ s$ and $F (s_{n}) ⇀ z$ , then $z = F (s)$ , where weak convergence in $R$ means convergence under every bounded real-linear functional. For every $α > 0$ , $x_{0}, y \in R$ , its Tikhonov functional

J_{α, y, x_{0}} (s) = \frac{1}{2} ∣ F (s) - y ∣^{2} + \frac{α}{2} ∣ s - x_{0} ∣^{2}

has a global minimizer on $R$ .

Example references

Nonlinear forward maps on open real Hilbert domains

Weakly closed domain and sequential graph closure

The nonlinear Tikhonov functional

Global minimizer existence

Measure spaces

Banach-valued simple integrals

Bochner integrability

Real Hilbert spaces

Weak sequence convergence

Facts & Assumptions

Given: The ultrafilter lemma, DC, and HB; the finite set $Ω = {0, 1}$ with $μ (A) = ∣ A ∣ / 2$ ; the maps $g_{s} (0) = s$ , $g_{s} (1) = s^{2}$ ; and $X = Y = D = R$ with the usual real Hilbert structure.

[A1]

The ultrafilter lemma extends every filter to an ultrafilter (The ultrafilter extension principle (UL/BPI)).

[A2]

DC supplies an infinite sequence following an everywhere-extendible relation (The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain).

[A3]

HB is the real dominated-extension principle (The real dominated-extension principle as an additional hypothesis over ZF).

[F1]

A measure is zero on the empty set and countably additive on disjoint measurable sets (Measures on sigma-algebras); a measure space consists of a set, sigma-algebra, and measure (Measure spaces).

[F2]

A measurable simple Banach-valued function $g = \sum_{j} x_{j} 1_{A_{j}}$ with finite-measure nonzero level sets is integrable and has integral $\int g d μ = \sum_{j} μ (A_{j}) x_{j}$ (Banach-valued simple function and integral).

[F3]

An integrable simple Banach-valued function is Bochner integrable, with its Bochner integral equal to the simple integral (Bochner-integrable function, The Banach-valued simple integral is well defined).

[F4]

The real numbers with their standard inner product form a real Hilbert space (Real and complex inner-product spaces and their induced length, The reals are complete, Hilbert space).

[F5]

Weak convergence in a normed space is convergence under every bounded real-linear functional (Weak convergence of nets and sequences).

[F6]

The open-domain nonlinear forward-map interface permits any map $F : U \to Y$ on a nonempty norm-open set, without assuming linearity (Nonlinear forward maps between Hilbert spaces).

[F7]

Under the assumptions of [A1]–[A3], a real-Hilbert weakly closed forward map on a nonempty weakly closed domain has a global minimizer for every positive parameter and datum (Existence of a global nonlinear Tikhonov minimizer).

[F8]

The Tikhonov functional is $\frac{1}{2} ∥ F (s) - y ∥^{2} + \frac{α}{2} ∥ s - x_{0} ∥^{2}$ on the domain (The nonlinear Tikhonov functional).

[F9]

DC implies countable choice, the principle that every countable family of nonempty sets admits a choice function (Dependent choice implies countable choice, The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F10]

A weakly closed forward map has a weakly closed domain and sequentially weak-to-weak closed graph (Weakly closed nonlinear forward maps on Hilbert spaces).

Verification

1.1givenF1F2F3algebra

The assignment $μ (A) = ∣ A ∣ / 2$ is a measure: it is zero on $\emptyset$ , and any disjoint sequence of subsets of the two-point set has at most two nonempty members, so cardinalities, hence measures, add over the union. For $s \notin {0, 1}$ , the two singleton level sets of $g_{s}$ have distinct nonzero coefficients $s, s^{2}$ and measure $1 / 2$ each. For $s = 0$ , use the empty representation of the zero function; for $s = 1$ , use $g_{1} = 1_{Ω}$ . In all cases [F2] gives an integrable simple function with integral $\frac{1}{2} s + \frac{1}{2} s^{2}$ , and the constant approximating sequence in [F3] gives the same Bochner integral. Since $F (1) = 1$ , $F (- 1) = 0$ , and $F (0) = 0$ , it fails additivity and is nonlinear.

1.2givenF5algebra

The domain $D = R$ is the entire weak topology space and is therefore weakly closed. If $s_{n} ⇀ s$ , [F5] applied to the identity functional gives $s_{n} \to s$ in the usual real topology. The factorization $F (s_{n}) - F (s) = \frac{1}{2} (s_{n} - s) (1 + s_{n} + s)$ shows $F (s_{n}) \to F (s)$ , because convergence of $s_{n}$ makes the second factor bounded. If also $F (s_{n}) ⇀ z$ , [F5] gives $F (s_{n}) \to z$ ; uniqueness of real limits yields $z = F (s)$ . Thus the graph is sequentially weak-to-weak closed.

2.1A1A2A3F4F6F7F8F9F10step 1.2construct∎

The domain $U = R$ is norm-open and nonempty, so [F6] confirms that this is also a forward map in the batch-25 open-domain convention. The graph and domain conditions just verified satisfy [F10]; with the real-Hilbert structure [F4] and axioms [A1]–[A3], they meet the hypotheses of [F7]. The DC-to-countable-choice consequence [F9] is the selection strength used in that theorem's proof. By [F8] this is the stated functional, so for every $α > 0$ , $x_{0}, y \in R$ it attains a global minimum.

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