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Compatible Banach pairs and their integrals

Statement

Fix a common scalar field $K \in {R, C}$ . Let $B_{1}, B_{2}$ be Banach spaces over $K$ , let $A$ be a Hausdorff topological vector space over $K$ , and let $j_{i} : B_{i} \to A$ be continuous injective $K$ -linear maps. The compatible-pair space $V = {(u, v) \in B_{1} \times B_{2} : j_{1} u = j_{2} v}$ , with norm $∥ (u, v) ∥_{V} = ∥ u ∥_{B_{1}} + ∥ v ∥_{B_{2}}$ , is Banach.

Let $(S, Σ, μ)$ be any measure space. If $f : S \to V$ , $f (r) = (u_{r}, v_{r})$ , is strongly measurable and $\int_{S} ∥ f (r) ∥_{V} d μ (r) < \infty$ , its Bochner integral $I = \int_{S} f d μ$ exists in $V$ . The coordinate maps $r \mapsto u_{r}$ and $r \mapsto v_{r}$ are Bochner integrable in $B_{1}$ and $B_{2}$ , respectively, and $j_{1} (\int_{S} u_{r} d μ (r)) = j_{2} (\int_{S} v_{r} d μ (r))$ . Scalar norm integrals use the measurable-representative convention supplied by strong simple approximation.

This criterion supplies compatibility from an actual closed Banach space, rather than assuming the existence or identification of a limit. No integral with values in the ambient topological vector space is required.

Statement references

Banach space

A closed subspace of a Banach space is Banach

Bochner integrability criterion

Bounded linear maps commute with Bochner integration

Measure spaces

Topological vector spaces over the real and complex fields

Strongly measurable Banach-valued function

Bochner-integrable function

Finite products of Banach spaces are Banach

Continuity of a map of topological spaces at a point and globally

Hausdorff space: distinct points have disjoint open neighbourhoods; every metrizable space is Hausdorff and the indiscrete topology on two points is not

The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement

Facts & Assumptions

[F1]

Banach spaces are complete for their norm metrics (Banach space). A finite product of Banach spaces is Banach for the sum norm (Finite products of Banach spaces are Banach), and closed linear subspaces with the restricted norm are Banach (A closed subspace of a Banach space is Banach). The common scalar field and the Hausdorff topological vector space have the conventions of Topological vector spaces over the real and complex fields.

[F2]

The parameter domain is a measure space (Measure spaces). Strong measurability means almost-everywhere norm approximation by measurable Banach-valued simple functions (Strongly measurable Banach-valued function). Bochner integrability and the integral are defined by integrable simple approximation in the norm-integral sense (Bochner-integrable function).

[F3]

Strong measurability and an integrable norm imply Bochner integrability, with scalar norm integrals evaluated through measurable representatives supplied by the strong approximation (Bochner integrability criterion). Bounded linear maps between Banach spaces preserve Bochner integrability and commute with the integral (Bounded linear maps commute with Bochner integration).

[F4]

Distinct points of a Hausdorff space have disjoint open neighborhoods (Hausdorff space: distinct points have disjoint open neighbourhoods; every metrizable space is Hausdorff and the indiscrete topology on two points is not). Continuity at a point means that every open neighborhood of its image contains the image of some open neighborhood of that point (Continuity of a map of topological spaces at a point and globally). For a norm metric, a neighborhood of a point contains a ball of positive radius; a set is open if each of its points has such a ball inside it, and a set is closed if its complement is open (The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement).

Proof

Given: Banach spaces and continuous injective linear maps into a common Hausdorff topological vector space, all over the fixed field $K$ ; a measure space $(S, Σ, μ)$ and a strongly measurable $f : S \to V$ with integrable $V$ norm.

Goal: Prove completeness and preservation of compatibility under the asserted integrable parameter family.

1.1F1given

Applying the finite-product theorem in [F1] to $B_{1}, B_{2}$ shows that $B_{1} \times B_{2}$ is Banach for the norm $∥ (u, v) ∥ = ∥ u ∥_{B_{1}} + ∥ v ∥_{B_{2}}$ . The set $V$ is a linear subspace because $j_{1}, j_{2}$ are linear.

2.1F1F4step 1.1

Fix $(u, v) \notin V$ , so $j_{1} u \neq j_{2} v$ . Hausdorff separation in [F4] gives disjoint open sets $O_{1}, O_{2}$ containing these two images. By continuity in [F4], there are open neighborhoods $N_{1} ∋ u$ in $B_{1}$ and $N_{2} ∋ v$ in $B_{2}$ with $j_{i} (N_{i}) \subseteq O_{i}$ . The norm-metric convention in [F4] gives radii $a, b > 0$ such that ${u^{'} : ∥ u^{'} - u ∥_{B_{1}} < a} \subseteq N_{1}$ and ${v^{'} : ∥ v^{'} - v ∥_{B_{2}} < b} \subseteq N_{2}$ . Set $δ = \min {a, b} > 0$ . If $∥ u^{'} - u ∥_{B_{1}} + ∥ v^{'} - v ∥_{B_{2}} < δ$ , then $j_{1} u^{'} \in O_{1}$ and $j_{2} v^{'} \in O_{2}$ , hence $j_{1} u^{'} \neq j_{2} v^{'}$ . Thus every point of $(B_{1} \times B_{2}) ∖ V$ has a sum-norm ball contained in that complement. By [F4] the complement is open and $V$ is closed. Applying the closed-subspace theorem in [F1] to step 1.1 proves that the restricted sum norm makes $V$ Banach. This uses only the two radii for the fixed point, with no countable selection.

3.1F2F3step 2.1∎

By [F2] and [F3], $f$ is Bochner integrable into $V$ , so $I = \int_{S} f d μ$ belongs to $V$ . Its coordinate projections $p_{i} : V \to B_{i}$ are linear and satisfy $∥ p_{i} z ∥_{B_{i}} \leq ∥ z ∥_{V}$ . Applying [F3] to $p_{i}$ shows that each coordinate function is Bochner integrable and $p_{i} I = \int_{S} p_{i} f d μ$ . Since $I \in V$ , $j_{1} (p_{1} I) = j_{2} (p_{2} I)$ , which is the asserted equality of images of the coordinate integrals. This argument does not require an integral with values in $A$ itself.

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