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Infinite-valued no-sinks Hahn–Banach sandwich and conditional offsets

Statement

Assume the Axiom of Choice. Let $V$ be a real vector space and $p : V \to R \cup {+ \infty}$ satisfy $p (0) = 0$ , $p (t x) = t p (x)$ for every $t > 0$ , and $p (x + y) \leq p (x) + p (y)$ in extended-real arithmetic. Assume no sinks: $p (x) < + \infty$ if and only if $p (- x) < + \infty$ . Put $W : = {x \in V : p (x) < + \infty}$ . Then $W$ is a linear subspace and there exists a real-linear $L : V \to R$ such that

$- p (- x) \leq L (x) \leq p (x) (x \in V),$

with $- \infty \leq r \leq + \infty$ interpreted in the usual extended order. The result is valid in arbitrary dimension and allows $p = + \infty$ outside $W$ .

For the conditional offset application, let $V = V_{G}$ from Type-offset quotient vector space, let $F : G \times G \to R \cup {+ \infty}$ satisfy $p ([e_{Γ}] - [e_{Γ^{'}}]) = F (Γ, Γ^{'})$ , and define $B (Γ) : = L ([e_{Γ}])$ . Then $B (Γ \times Γ^{'}) = B (Γ) + B (Γ^{'})$ , $B (t Γ) = t B (Γ)$ for $t > 0$ , and

$- F (Γ^{'}, Γ) \leq B (Γ) - B (Γ^{'}) \leq F (Γ, Γ^{'}) .$

For each $Γ \in G$ , let $X_{Γ}$ be a state set and $σ_{Γ} : X_{Γ} \to R$ a baseline state function. If every admitted transition $(Γ, x) \to (Γ^{'}, y)$ , with $x \in X_{Γ}$ and $y \in X_{Γ^{'}}$ , satisfies $σ_{Γ^{'}} (y) - σ_{Γ} (x) \geq F (Γ, Γ^{'})$ , then $σ_{Γ} (x) + B (Γ)$ is nondecreasing across those transitions. The existence or properties of such a cost $p$ , including no-sinks, are hypotheses and are not proved by this theorem.

Facts & Assumptions

[A1]

The Axiom of Choice: adopt the Axiom of Choice for the Hahn–Banach and basis-extension steps.

[F1]

Vector space over a field and Linear subspace of a vector space: $V$ is a real vector space, and a subset closed under addition and all scalar multiples is a linear subspace.

[F2]

A sublinear functional on a real vector space: a real-valued sublinear functional is positively homogeneous and subadditive.

[F3]

Hahn-Banach dominated extension theorem for real vector spaces: for real vector spaces, a subspace, a real-valued sublinear functional and a linear functional dominated on that subspace, there is a dominated linear extension to the whole space.

[F4]

Complete ordered field (least-upper-bound property): the real field used in the one-step dominated-extension argument has the least-upper-bound property.

[F5]

Every vector space has a basis and Zorn's lemma gives a basis between any linearly independent set and any spanning set containing it: if $L \subseteq S \subseteq V$ with $L$ independent and $span (S) = V$ , there is a basis $B$ of $V$ with $L \subseteq B \subseteq S$ : under the adopted choice strength, a vector space has a basis and an independent subset extends to a basis of the ambient space.

[F6]

Linear independence: a finite list $v : n \to V$ is independent when $\sum_{i < n} λ_{i} v_{i} = 0_{V}$ forces every $λ_{i} = 0_{F}$ , and a subset $S \subseteq V$ is independent when every injective finite list into $S$ is independent and Basis of a vector space: a linearly independent spanning subset; and ordered basis: an injective finite list whose image is a basis: a basis is an independent spanning set; spanning gives a finite expression for each vector, and independence makes its finite coefficient map unique.

[F7]

Type-offset quotient vector space and Type-offset quotient is a real vector space: the quotient is a real vector space with the declared product and positive-scaling relations.

[F8]

The extended real line $\overline{R} = R \cup {- \infty, + \infty}$ , its order, and the arithmetic that is left undefined: the order and negation on $R \cup {\pm \infty}$ are extended as defined there; for values in $R \cup {+ \infty}$ , addition is the real sum when both are real and is $+ \infty$ when either is $+ \infty$ , while multiplication by $t > 0$ sends $+ \infty$ to $+ \infty$ . These cases avoid the undefined extended-real operations listed in that definition.

[F9]

Linear functionals and the algebraic dual $V^{*} = L (V, F)$ : a real-linear functional $L : V \to R$ is a linear map, so $L (x + y) = L (x) + L (y)$ and $L (t x) = t L (x)$ for $x, y \in V$ and $t \in R$ .

[F10]

The Axiom Schema of Separation: for each formula $φ$ , $\forall \overset{ˉ}{p} \forall x \exists y \forall z (z \in y \leftrightarrow (z \in x \land φ (z, \overset{ˉ}{p})))$ : since $p$ is a function on the set $V$ and $+ \infty$ is in the extended-real codomain, Separation forms the subset $W = {x \in V : p (x) < + \infty}$ as a set.

Proof

Given: [A1]–[F10] and the hypotheses on $p$ .

1.1F1F2F8F10algebra

By [F10], $W$ is a set. It contains $0$ because $p (0) = 0$ . If $x, y \in W$ , subadditivity gives $p (x + y) \leq p (x) + p (y) < + \infty$ , so $x + y \in W$ by the extended-real conventions in [F8]. For $t > 0$ , homogeneity gives $p (t x) = t p (x) < + \infty$ . If $t < 0$ , put $s : = - t > 0$ ; no-sinks gives $- x \in W$ , and positive homogeneity applied to $s (- x) = t x$ puts $t x$ in $W$ . Also $0 x = 0 \in W$ . Thus $W$ is a linear subspace. The restriction $p ∣_{W}$ is finite-valued, positively homogeneous and subadditive, so it is a real-valued sublinear functional.

2.1A1F3F4F8F9step 1.1

Let $M = {0} \subseteq W$ and let $f_{0} (0) = 0$ . This is linear and satisfies $f_{0} (0) = 0 = p (0)$ . The real ordered field is complete by [F4], as used in the one-step interval argument of the published theorem. Apply [F3] to $p ∣_{W}$ and $f_{0}$ ; it gives a linear $ℓ : W \to R$ with $ℓ (x) \leq p (x)$ for every $x \in W$ . Applying this at $- x \in W$ yields $- ℓ (x) = ℓ (- x) \leq p (- x)$ , so $- p (- x) \leq ℓ (x) \leq p (x)$ on $W$ .

3.1A1F5F6F9step 2.1

By [F5], choose a basis $B_{W}$ of $W$ and extend its independent set to a basis $B_{V}$ of $V$ . Assign value $ℓ (b)$ to $b \in B_{W}$ and $0$ to $b \in B_{V} ∖ B_{W}$ . For any $v \in V$ , [F6] gives a finite basis expression; define $L (v)$ as the same finite linear combination of assigned real values. If two expressions represented $v$ , subtracting and collecting equal basis vectors would give a finite zero combination; independence in [F6] forces every collected coefficient to be zero. Thus $L (v)$ is well-defined. Concatenating finite expressions proves additivity and scalar homogeneity, so $L$ is linear in the sense of [F9]; the basis expression of any $w \in W$ uses only $B_{W}$ , so $L ∣_{W} = ℓ$ .

4.1F8step 1.1step 2.1step 3.1

For $x \in W$ , step 3.1 and step 2.1 give $- p (- x) \leq L (x) \leq p (x)$ . If $x \notin W$ , then $p (x) = + \infty$ by the definition of $W$ ; no-sinks implies $p (- x) = + \infty$ as well, so by [F8] the extended bounds reduce to $- \infty \leq L (x) \leq + \infty$ . This proves the global sandwich.

5.1F7F9step 4.1algebra∎

In the type-offset application, linearity of $L$ in the sense of [F9] and the quotient relations in [F7] give $B (Γ \times Γ^{'}) = B (Γ) + B (Γ^{'})$ and $B (t Γ) = t B (Γ)$ . Apply the sandwich to $e_{Γ} - e_{Γ^{'}}$ and its negative, using the assumed values of $p$ , to obtain the two displayed bounds for $B (Γ) - B (Γ^{'})$ . For an admitted transition, its real baseline change is at least $F (Γ, Γ^{'})$ , so that cost value must be finite. The corrected change equals $[σ_{Γ^{'}} (y) - σ_{Γ} (x)] + B (Γ^{'}) - B (Γ) = [σ_{Γ^{'}} (y) - σ_{Γ} (x)] - L (e_{Γ} - e_{Γ^{'}})$ . The assumed lower bound on the baseline change and the upper sandwich bound $L (e_{Γ} - e_{Γ^{'}}) \leq F (Γ, Γ^{'})$ make this at least $F (Γ, Γ^{'}) - F (Γ, Γ^{'}) = 0$ . This proves only the stated conditional monotonicity.

Remarks

The real-valued Hahn–Banach theorem is applied only on the finite domain $W$ ; the arbitrary-dimensional extension to $V$ uses a basis and its extension under Choice. When $x \notin W$ , both sandwich endpoints are infinite by no-sinks. Neither the finite-dimensional strict-extension variant nor a catalyst-chain cost construction is asserted.

Infinite-valued no-sinks Hahn–Banach sandwich and conditional offsets

Statement

Facts & Assumptions

Proof

Remarks

Depends on

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