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The Cameron–Martin unit ball is compact in the Banach norm

Statement

Assume AC. Let $μ$ be a centered tight Borel Gaussian law on a real separable Banach space $B$ , and let $H \subset B$ be its covariance RKHS. The closed Hilbert unit ball $H_{1} = {h \in H : ∥ h ∥_{H} \leq 1}$ is compact in the Banach norm.

Facts & Assumptions

[A1]

AC is inherited from the Gaussian translation and tightness suppliers. Under AC, the ultrafilter lemma follows from The ultrafilter lemma, from the Axiom of Choice: every filter extends to an ultrafilter, and DC and countable choice follow from AC implies DC implies countable choice.

[A2]

Under AC, the real dominated-extension theorem supplies the Hahn–Banach principle HB used by the dual-norming interface (Hahn-Banach dominated extension theorem for real vector spaces, The real dominated-extension principle as an additional hypothesis over ZF).

[F1]

For $h \in H$ , the translated law has density $D_{h} = \exp (z_{h} - ∥ h ∥_{H}^{2} / 2)$ , where $z_{h}$ is centered normal of variance $∥ h ∥_{H}^{2}$ (Cameron–Martin translation formula for a separable Banach Gaussian, Standard normal and normal laws).

[F2]

A Borel probability on the separable Banach space is tight (Every borel probability on a polish space is tight, Prokhorov tightness theorem on polish spaces).

[F3]

Under the ultrafilter lemma, DC and HB, a Banach space is reflexive exactly when every bounded sequence has a weakly convergent subsequence; under countable choice, a Hilbert space is reflexive (Reflexivity is equivalent to weak subsequential compactness of bounded sequences, Hilbert spaces are reflexive by Riesz representation). The needed choice hypotheses follow from [A1] and [A2].

[F4]

The canonical embedding satisfies $ℓ (i h) = ⟨ h, [ℓ] ⟩_{H}$ , and continuous linear functionals separate points (Covariance operator and reproducing-kernel Hilbert space of a Banach Gaussian, Relative dual norming, point separation, and recovery of the norm).

[F5]

Cauchy–Schwarz applies to the square-integrable density and event indicator (Cauchy-Schwarz for random variables).

Proof

Given: AC and a centered tight Borel Gaussian law $μ$ on $B$ , its covariance RKHS $H$ , and $H_{1}$ .

1.1A1F1F2F5givenalgebra

If $h \in H_{1}$ , write $v = ∥ h ∥_{H}^{2} \leq 1$ . If $v = 0$ , then $z_{h} = 0$ almost surely and $D_{h} = 1$ . If $v > 0$ , completing the square in the normal density of $z_{h}$ gives $E e^{2 z_{h}} = e^{2 v}$ . In both cases [F1] yields $\int D_{h}^{2} d μ = e^{v} \leq e$ . Choose compact $K \subset B$ with $μ (K^{c}) < 1 / (16 e)$ using [F2]. Cauchy–Schwarz [F5] then gives $μ_{h} (K^{c}) = \int_{K^{c}} D_{h} d μ < 1 / 4$ , uniformly for all $h \in H_{1}$ , while $μ (K) > 3 / 4$ . Hence $μ (K \cap (K - h)) \geq μ (K) + μ_{h} (K) - 1 > 0$ . For each $h \in H_{1}$ there is $x \in K$ with $x + h \in K$ , so $h \in K - K$ . The difference $K - K$ is compact; therefore $H_{1}$ is relatively compact in $B$ .

2.1A1A2F3F4step 1.1algebra∎

Let $(h_{n})$ be any sequence in $H_{1}$ . By [F3], pass to a subsequence weakly converging in $H$ to some $h \in H_{1}$ . Relative compactness from step 1.1 gives a further subsequence with $i h_{n} \to y$ in $B$ . For every $ℓ \in B^{*}$ , [F4] gives $ℓ (y) = \lim_{n} ⟨ h_{n}, [ℓ] ⟩_{H} = ⟨ h, [ℓ] ⟩_{H} = ℓ (i h)$ . By [F4], $y = i h$ . Thus every sequence in $H_{1}$ has a norm-convergent subsequence whose limit lies in $H_{1}$ ; in the metric space $B$ , this is compactness.

Depends on

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Sources

A. Lunardi, M. Miranda, and D. Pallara, Infinite Dimensional Analysis (standard reference, not scraped)
M. Hairer, An Introduction to Stochastic PDEs (standard reference, not scraped)