DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)

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.

Concentration function of a Banach Gaussian prior

Definition

Assume AC and countable choice. Let $μ$ be a centered tight Borel Gaussian law on a real separable Banach space $B$ , with covariance RKHS $H$ , and let $w_{0} \in {\overline{H}}^{B}$ . For $ε > 0$ , write $ε B^{\circ} = {x \in B : ∥ x ∥_{B} < ε}$ and define the concentration function $ϕ_{w_{0}} (ε) = \inf_{\begin{array}{c} h \in H \\ ∥ h - w_{0} ∥_{B} < ε \end{array}} {∥ h ∥_{H}^{2} - \log μ (ε B^{\circ})} .$ The feasible set is nonempty because $w_{0}$ lies in the Banach-norm closure of $H$ . Also $μ (ε B^{\circ}) > 0$ because $0 \in supp μ$ ; thus the logarithm is finite. The value is finite and nonnegative. The support assertion used here is The support of a Banach Gaussian is the closure of its RKHS.

Depends on

Used by

Dependency tree · two levels

30 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

A. W. van der Vaart and J. H. van Zanten, Rates of contraction of posterior distributions based on Gaussian process priors (2008) (standard reference, not scraped)
A. Lunardi, M. Miranda, and D. Pallara, Infinite Dimensional Analysis (standard reference, not scraped)