L vector spaces and L fields

We construct in ZFC an L topological vector space -- a topological vector space that is an L space -- and an L field -- a topological field that is an L space. This generalizes results in [5] and [8].Comment: It has been accepted for publication in SCIENCE CHINA Mathematic

Holonomic Spaces

A holonomic space $(V,H,L)$ is a normed vector space, $V$, a subgroup, $H$, of $Aut(V, \|\cdot\|)$ and a group-norm, $L$, with a convexity property. We prove that with the metric $d_L(u,v)=\inf_{a\in H}\{\sqrt{L^2(a)+\|u-av\|^2}\}$, $V$ is a metric space which is locally isometric to a Euclidean ball. Given a Sasaki-type metric on a vector bundle $E$ over a Riemannian manifold, we prove that the triplet $(E_p,Hol_p,L_p)$ is a holonomic space, where $Hol_p$ is the holonomy group and $L_p$ is the length norm defined within. The topology on $Hol_p$ given by the $L_p$ is finer than the subspace topology while still preserving many desirable properties. Using these notions, we introduce the notion of holonomy radius for a Riemannian manifold and prove it is positive. These results are applicable to the Gromov-Hausdorff convergence of Riemannian manifolds.Comment: 17 page

Quotient spaces of boundedly rational types

By identifying types whose low-order beliefs – up to level li – about the state of nature coincide, we obtain quotient type spaces that are typically smaller than the original ones, preserve basic topological properties, and allow standard equilibrium analysis even under bounded reasoning. Our Bayesian Nash (li; l-i)-equilibria capture players’ inability to distinguish types belonging to the same equivalence class. The case with uncertainty about the vector of levels (li; l-i) is also analyzed. Two examples illustrate the constructions.Incomplete-information games, high-order reasoning, type space, quotient space, hierarchies of beliefs, bounded rationality