A Proof of the Barsotti-Chevalley Theorem on Algebraic Groups

Another proof of Banaschewski's surjection theorem

We present a new proof of Banaschewski's theorem stating that the completion lift of a uniform surjection is a surjection. The new procedure allows to extend the fact (and, similarly, the related theorem on closed uniform sublocales of complete uniform frames) to quasi-uniformities ("not necessarily symmetric uniformities"). Further, we show how a (regular) Cauchy point on a closed uniform subl...

Some Results on TVS-cone Normed Spaces and Algebraic Cone Metric Spaces

In this paper we introduce the cone bounded linear mapping and demonstrate a proof to show that the cone norm is continuous. Among other things, we prove the open mapping theorem and the closed graph theorem in TVS-cone normed spaces. We also show that under some restrictions on the cone, two cone norms are equivalent if and only if the topologies induced by them are the same. In the sequel, we...

Two Proofs of a Theorem on Algebraic Groups

A Modern Proof of Chevalley’s Theorem on Algebraic Groups

Let k be a field, and let G be an algebraic group over k, by which we mean a connected smooth k-group scheme (not necessarily affine). Recall that such a G is automatically separated, finite type, and geometrically integral over k [4, Exp VIA, 0.3, 2.1.2, 2.4]. The most important classes of algebraic groups are the affine algebraic groups (also called linear algebraic groups, since affine algeb...

Some Aspects of the Structure and Representation Theory of Algebraic Groups

This expository paper discusses a few of the results concerning the structure theory and representation theory of algebraic groups over a fixed algebraically closed field K, such as the linearization of affine groups, the Jordan-Chevalley decomposition, and the Lie-Kolchin theorem.