A characterization of Walrasian economies of infinity dimension

We consider a pure exchange economy, where agent's consumption spaces are Banach spaces, goods are contingent in time of states of the world, the utility function of each agent is not necessarily a separable function, but increasing, quasiconcave, and twice Frechet differentiable over the consumption space. We characterize the set of walrasian equilibria, by the social weight that support each walrasian equilibria. Using technical of the functional analysis, we characterize this set as a Banach manifold and in the next sections we focuses on singularities.

Limited operators and differentiability

We characterize the limited operators by differentiability of convex continuous functions. Given Banach spaces $Y$ and $X$ and a linear continuous operator $T: Y \longrightarrow X$, we prove that $T$ is a limited operator if and only if, for every convex continuous function $f: X \longrightarrow \R$ and every point $y\in Y$, $f\circ T$ is Fr\'echet differentiable at $y\in Y$ whenever $f$ is G\^ateaux differentiable at $T(y)\in X$

Sobolev functions on varifolds

This paper introduces first order Sobolev spaces on certain rectifiable varifolds. These complete locally convex spaces are contained in the generally nonlinear class of generalised weakly differentiable functions and share key functional analytic properties with their Euclidean counterparts. Assuming the varifold to satisfy a uniform lower density bound and a dimensionally critical summability condition on its mean curvature, the following statements hold. Firstly, continuous and compact embeddings of Sobolev spaces into Lebesgue spaces and spaces of continuous functions are available. Secondly, the geodesic distance associated to the varifold is a continuous, not necessarily H\"older continuous Sobolev function with bounded derivative. Thirdly, if the varifold additionally has bounded mean curvature and finite measure, the present Sobolev spaces are isomorphic to those previously available for finite Radon measures yielding many new results for those classes as well. Suitable versions of the embedding results obtained for Sobolev functions hold in the larger class of generalised weakly differentiable functions.Comment: Version initially accepted by Proc. Lond. Math. Soc. (3). The final printed version will be different. 55 pages, no figure