Funtional analysis lecture notes for 18 mit mathematics. On orthocomplemented subspaces in padic banach spaces. Thanks to abhinav kumar for providing the corrections incorporated into this version. Riesz lemma is an elementary result often an adequate substitute in banach spaces for the lack of sharper. Completely continuous endomorphisms of padic banach spaces.
Bhargav bhatt perfectoid spaces are a class of algebrogeometric objects living in the realm of p adic geometry that were introduced by peter scholze sch12 in his. Request pdf on orthocomplemented subspaces in padic banach spaces for linear subspaces of finitedimensional normed spaces over k, where k is a nonarchimedean complete valued field which is. On orthocomplemented subspaces in padic banach spaces by albert kubzdela institute of civil engineering, poznatl university of technology, ui. Let v be a finitedimensional normed vector space over k. The purpose of the current paper is to introduce some new methods for studying the spaces. The author begins with a discussion of weak topologies, weak compactness and isomorphisms of banach spaces before proceeding to the more detailed study of particular spaces. To this end, we introduce the category of spectral banach spaces, which are p adic banach spaces with a. Completely continuous endomorphisms of padic banach spaces by jeanpierre serre publ. This is a correspondence between unitary continuous admissible representaitons of gl2q pvaluedinpadic banach spaces, and unitary continuous representations of d. Isometries in banach spaces download ebook pdf, epub. However, we will not need in nite dimensional spaces here. Peter scholzes lectures on padic geometry, fall 2014 last updated on 29 jan. Escassut, ultrametric banach algebras, world scientific, 2003.
Padic banach spaces and families of modular forms robert f. Completely continuous endomorphisms of adic banach spaces. The number p is a real number with 1 p adic banach spaces and families of modular forms robert f. All vector spaces are assumed to be over the eld f. Given a linear algebraic group g defined over a number field, emerton has defined a sequence of padic banach spaces these spaces are representations of the group of adele points of g, and may be regarded as spaces of padic automorphic forms. Introduction in the construction of eigenvarieties, the space of overconvergent modular forms arises as an in nitedimensional padic banach space. The padic numbers are a counterintuitive arithmetic system that was discovered by the german mathematician kurt hensel in about 1899. Let v be a banach space, and let u v be a closed vector subspace. Weak and strong ccompactness in non archimedean banach spaces. On irreducible representations of compact padic analytic. Introduction let pbe a prime, cp the completion of an algebraic closure of the p adicnumbers qp and ka. In this case, the representations of g which took part in the correspondence were banach spaces with a ginvariant norm. Thanks to arthur ogus and jay pottharst for helpful comments.
Piotrowo 5, 618 poznah, poland communicated by prof. Analytic padic banach spaces and the fundamental lemma of. Introduction in the construction of eigenvarieties, the space of overconvergent modular forms arises as an in nitedimensional p adic banach space. Complemented subspaces of padic second dual banach spaces article pdf available in international journal of mathematics and mathematical sciences 183 january 1995 with 26 reads. Berkovich, spectral theory and analytic geometry over nonarchimedean fields, ams, 1990 the above conditions are in corollary 9. It is online only to give the audience of our lecture easy access to it. It showcases research results in functional analysis. Introduction let pbe a prime, cp the completion of an algebraic closure of the padicnumbers qp and ka.
Schikhof mathematical institute, university of nijmegen, toernooiveld, 6525 ed nijmegen. Abstractly, banach spaces are less convenient than hilbert spaces, but still su ciently simple so many important properties hold. In mathematics, p adic analysis is a branch of number theory that deals with the mathematical analysis of functions of p adic numbers the theory of complexvalued numerical functions on the p adic numbers is part of the theory of locally compact groups. Throughout, f will denote either the real line r or the complex plane c. There is a beautiful theory of representations over such and so there is no serious problem. The book is intended to be used with graduate courses in banach space theory, so the prerequisites are a background in functional, complex and real analysis. Mathematically it is natural to work over the padic. Springer at the meeting of november 29, 2004 abstract. The closely related adeles and ideles were introduced in the 1930s by claude chevalley. A very basic fact about banach spaces is the following so called open mapping theorem. Exercise 1 sequence spaces p are banach spaces given a sequence x 1. Theres a lot more where this came from, but we wont need the rest of it just yet.
Proceedings of the international conference held in trento, italy, may 29june 2, 1989. The u p operator acts on this space as a compact operator, and cuts out a nitedimensional locus on which it acts with a bounded slope. A padic jacquetlanglands correspondence abstract in this paper, we construct a candidate padic jacquetlanglands correspondence. In lectures 1215, we study shtukas with one leg, and their connection to pdivisible groups and padic hodge theory. Peter scholzes lectures on padic geometry, fall 2014.
A presentation of results in padic banach spaces, spaces over fields with an infinite rank valuation, frechet and locally convex spaces with schauder bases, function spaces, padic harmonic analysis, and related areas. Schikhof mathematical institute, university ofnijmegen, toernooiveld, 6525 ed nijmegen, the netherlands communicated by prof. Completely continuous endomorphisms of p adic banach spaces by jeanpierre serre publ. A padic shimura isomorphism and padic periods of modular forms, contemporary mathematics 165. P adic banach spaces and families of modular forms. Throughout this course k is a finite extension of the field qp of padic numbers, denotes the absolute. In part a, which we entitle families of banach spaces, we show how serres p adicbanachfredholmriesz theory s works in a family, i. It is the application of padic analysis to quantum mechanics. Several standard results true in greater generality have simpler proofs for banach spaces. Springer at the meeting of september 26,1994 abstract in the set up of nonarchimedean functional analysis, banach spaces as well as.
A p adic shimura isomorphism and p adic periods of modular forms. Springer at the meeting of september 26,1994 abstract. The usual meaning taken for p adic analysis is the theory of p adic valued functions on spaces of interest. On emertons padic banach spaces oxford academic journals. Padic quantum mechanics is a relatively recent approach to understanding the nature of fundamental physics. Given a linear algebraic group g defined over a number field, emerton 10 has defined a sequence of padic banach spaces. A perfect duality between padic banach spaces and compactoids by w. Note that there is no natural involution in the padic case. It showcases research results in functional analysis over nonarchimedean valued complete fields. Coleman this paper is dedicated to bernard dwork who has been a friend and an inspiration for many years. The complementation property of linfinity in p adic banach spaces.
Such a correspondence has been shown for g gl2 when the. Banachhecke algebras and adic galois representations. Nonarchimedean banach spaces over trivially valued fields. This example corresponds to a point in the world of perfectoid kspaces. To this end, we introduce the category of spectral banach spaces, which are padic banach spaces with a c palgebra of analytic functions, and the subcategory of e. But as there are no hilbert spaces we have to work in the category of banach spaces over the.
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