用泛函观点看LEBESGUE积分

从泛函分析观点来看Lebesgue积分,使得Lebesgue积分可以用泛函分析最简单最基本的方法独立导出．基本做法是将Riemann对于区间[0,1]上的连续函数的积分看成连续函数空间C[0,1]上的连续线性泛函,再将它“自然”延拓到C[0,1]在积分范数意义下的完备化空间,而这个完备化空间正是Lebesgue可积函数空间L1[0,1]．     

一点看法

数学研究与评论1984年第四期刊登了刘证同志的“L~p-orthogonality in Banach Spaces” (以下简称文献[1])一文,此文给出了Banach空间的L~p一正交元的“存在性”及“唯一性”定理(文献[1]的定理1,2),这无疑证明了L~p-正交性(p>1)的提法对任何一个Banach空间都是非空的,为L~p-正交性的研究提供了理论依据。但就文献[1]中某些定理及推论的价值问题,笔者在此提出拙见,与刘证同志商榷。在文献[1]的推论4,6,7,9,定理8的条件下,p=2时,James和Day已经在文献[2],[3]给出了结论;     

关于Banach空间光滑性的两点注记

In this paper, We show that a real or complex Banach space is smooth at x₀(∈S(X)) iff, for any y∈S(X).(?)where X_x0,y= span {x₀, y} . For a real Banach space X, we obtain that X is F-differentia ble at x₀(∈S(X)) iff (?) Remark. The sufficient conditions of the conclusion have been proved by Yu Xintai [2].     

ON THE PRODUCT OF GTEAUX DIFFERENTIABILITY LOCALLY CONVEX SPACES

A locally convex space is said to be a Gateaux differentiability space (GDS) provided every continuous convex function defined on a nonempty convex open subset D of the space is densely Gateaux differentiable in .D.This paper shows that the product of a GDS and a family of separable Prechet spaces is a GDS,and that the product of a GDS and an arbitrary locally convex space endowed with the weak topology is a GDS.     

ANoteonWeaklyCompactSetsinBanachSpace

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有限维赋范线性空间光滑曲线的求长

本文给出了有限维赋范线性空间光滑曲线的求长公式。     

关于G-M成果研究的若干新动态Ⅰ——G-M型空间的若干品种

结合自己的工作，对Gowers-Maurey系列成果获Fields奖以来的研究的新动态作一综述。本是上篇，主要讨论含遗传不可分解空间在内的G－M型空间的若干品种。     

Yet on linear structures of norm-attaining functionals on Asplund spaces

In this paper, we show that if an Asplund space X is either a Banach lattice or a quotient space of C(K), then it can be equivalently renormed so that the set of norm-attaining functionals contains an infinite dimensional closed subspace of X^* if and only if X^* contains an infinite dimensional reflexive subspace, which gives a partial answer to a question of Bandyopadhyay and Godefroy.     

Banach空间的p— Asplund 伴随空间

我们称一个定义在Banach空间E上的连续凸函数f具有Frechet可微性质（FDP）,如果E上的每个实值凸函数g≤f均在E一个稠密的Gδ－子集上Frechet可微。本文主要证明了：对任何Banach空间E,均存在一个局部凸相容拓扑p使得1）（E,p)是Hausdorff局部凸空间;2） E上的每个范数连续具有FDP的凸函数均是p－连续的;3）每个p-连续的凸函数均具有FDP ;4）p等价某个范数拓扑当且仅不E是Asplund空间。