泛函分析基础命题的改进
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摘要
闭图象定理、开映射定理和等度连续定理(蕴涵一致有界原理即共鸣定理)是泛函分析理论的三大基本原理。泛函分析非常依赖Baire纲定理和Hahn-Banach延拓定理,但前者是一般拓扑学的命题而后者则正如W. Orlicz与我国学者夏道行所明确指出,是纯代数的。
经典的泛函分析三大基本原理有一个共同点就是所处理的映射只限于过分理想的线性算子,所以三大基本原理不适用于非线性映射。这就使三大基本原理的理论价值与应用都受到了很大的限制。于是人们推广和改进三大基本原理的艰苦努力持续了60多年之久,然而泛函分析的三大基本原理虽经60多年的推广与改进但就其本质而言至今仍停留在经典命题的理论水平上。本论文对泛函分析的三大基本原理作出了实质性的全面改进。事实上,本论文对三类很大的映射族分别建立了三大基本原理,这三类映射族每个都包含经典原理所处理的全部线性算子特别是每个类中的非线性映射不少于线性算子。从此泛函分析三大基本原理被提升为将线性分析作为特例的更具普遍意义的所谓泛线性分析的基本原理,从而使三大基本原理的理论价值与应用范围分别提升和扩大到新的高度。在新三大基本原理的基础上线性对偶理论已开拓成为泛线性对偶理论,特别是泛线性广义函数理论给出了一些颇具新意的结论,新的开映射定理也给出了一些颇有理论价值与现实意义的结果。
本研究的指导思想是:尽可能地反映现实生活。例如,线性算子作绝对的精确解剖:f(x + tz) = f(x) + tf(z), (?)x, z∈X, t∈C,这里绝对性指“(?)x, z∈X, (?)t∈C”,精确性则指x + tz中的系数1与t准确无误地出现在解剖结果上。但现实中的解剖现象往往不是如此的绝对精确。为反映这不理想的现实,我们界定了解剖映射:对每个x∈X, f(x + tu) = rf(x) + sf(u),其中u只限于在某一0点邻域中变动,t只限于|t|≤1,|r-1|与|s-t|则受到|t|的简单控制。因此解剖映射族是线性算子族的非常自然的大扩充,新的等度连续定理正是对解剖映射族建立的。又如线性算子具有绝对精确的可加性:f(x)+f(z) = f(x+z),这也是非常理想的。而f(x)+f(z) = f(u)其中的情形则相当普遍。正是由这个事实出发我们得到了新的开映射定理。新的闭图象定理也是以类似的思路得到的。
新三大基本原理的建立不仅使三大原理的理论水平得到明显提高,也扩大了三大原理的应用范围。对此本论文在最后一章给出了新三大基本原理在对偶理论、广义函数理论和Banach空间理论中的一些应用。
Closed graph theorem, open mapping theorem and equicontinuity theorem (imply-ing uniform boundedness principle, i.e., resonance theorem) are the three basic principlesof the theory of functional analysis. Functional analysis greatly depends on Baire cate-gory theorem and Hahn-Banach extension theorem. But the former is a proposition of thetheory of general topology and the latter, as W. Orlicz and a scholar of our country XiaDaoxing explicitly say, belongs to pure algebra.
There is a common property among classical three basic principles of functionalanalysis, that is, treated mappings are only linear operators which are too ideal. So thethree basic principles are not applicable to nonlinear mappings. This greatly makes thetheoretical value and applications of three basic principles limited. So there have beenmore than sixty years for our hard work to generalize and improve the three basic prin-ciples. However, the three basic principles are still as before with extensions and im-provements of more than sixty years. In this dissertation, we substantially give all-aroundimprovements of the three basic principles of functional analysis. In fact, we establish thenew three basic principles for three very large families of mappings respectively. Each ofthe three families of mappings includes all linear mappings handled in the classical prin-ciples. Especially, the number of nonlinear mappings in each of the three families are notfewer than that of linear mappings. Henceforth, the three basic principles of functionalanalysis are promoted to the basic principles of pan-linear analysis with general mean-ing which make linear analysis to be a special case. The theoretical value and field ofapplications of three basic principles are promoted and expanded to a new level. The lin-ear dual theory, based on the new three basic principles, has been extended to pan-lineardual theory. Especially, the theory of pan-linear distributions has given some very newresults, and new open mapping theorem also yields some results which have very strongtheoretical value and practical significance.
Our guided thought is to try to reflect real life. For example, each of linear operatorsmakes absolutely precise dissection: f(x + tz) = f(x) + tf(z), (?)x, z∈X, t∈C,where absoluteness means“(?) x, z∈X, (?)t∈C”and precision means that the coeffi- cients 1 and t accurately appear in the dissecting result. But dissecting phenomenons inreal life are not often so absolutely precise. To re?ect this unideal real life, we definitedissecting mappings: for each x∈X, f(x + tu) = rf(x) + sf(u), where u belongsto a neighborhood of 0, t satisfies |t|≤1, and |r - 1| and |s - t| are simply controlledby |t|. Hence, the family of dissecting mappings is a very natural large extension of thefamily of linear operators. The new equicontinuity theorem is just established for thefamily of dissecting mappings. And each of linear operators has absolutely precise addi-tivity: f(x) + f(z) = f(x + z). This is very ideal too. But f(x) + f(z) = f(u) where is fairly common. Following this fact, we obtain the new openmapping theorem. The new closed graph theorem is obtained as the same idea.
New three basic principles not only greatly promote the theoretical level, but alsoexpand the field of applications of three basic principles. In the last chapter, we give someapplications of new basic principles in dual theory, distributions and the theory of Banachspaces.
经典的泛函分析三大基本原理有一个共同点就是所处理的映射只限于过分理想的线性算子,所以三大基本原理不适用于非线性映射。这就使三大基本原理的理论价值与应用都受到了很大的限制。于是人们推广和改进三大基本原理的艰苦努力持续了60多年之久,然而泛函分析的三大基本原理虽经60多年的推广与改进但就其本质而言至今仍停留在经典命题的理论水平上。本论文对泛函分析的三大基本原理作出了实质性的全面改进。事实上,本论文对三类很大的映射族分别建立了三大基本原理,这三类映射族每个都包含经典原理所处理的全部线性算子特别是每个类中的非线性映射不少于线性算子。从此泛函分析三大基本原理被提升为将线性分析作为特例的更具普遍意义的所谓泛线性分析的基本原理,从而使三大基本原理的理论价值与应用范围分别提升和扩大到新的高度。在新三大基本原理的基础上线性对偶理论已开拓成为泛线性对偶理论,特别是泛线性广义函数理论给出了一些颇具新意的结论,新的开映射定理也给出了一些颇有理论价值与现实意义的结果。
本研究的指导思想是:尽可能地反映现实生活。例如,线性算子作绝对的精确解剖:f(x + tz) = f(x) + tf(z), (?)x, z∈X, t∈C,这里绝对性指“(?)x, z∈X, (?)t∈C”,精确性则指x + tz中的系数1与t准确无误地出现在解剖结果上。但现实中的解剖现象往往不是如此的绝对精确。为反映这不理想的现实,我们界定了解剖映射:对每个x∈X, f(x + tu) = rf(x) + sf(u),其中u只限于在某一0点邻域中变动,t只限于|t|≤1,|r-1|与|s-t|则受到|t|的简单控制。因此解剖映射族是线性算子族的非常自然的大扩充,新的等度连续定理正是对解剖映射族建立的。又如线性算子具有绝对精确的可加性:f(x)+f(z) = f(x+z),这也是非常理想的。而f(x)+f(z) = f(u)其中的情形则相当普遍。正是由这个事实出发我们得到了新的开映射定理。新的闭图象定理也是以类似的思路得到的。
新三大基本原理的建立不仅使三大原理的理论水平得到明显提高,也扩大了三大原理的应用范围。对此本论文在最后一章给出了新三大基本原理在对偶理论、广义函数理论和Banach空间理论中的一些应用。
Closed graph theorem, open mapping theorem and equicontinuity theorem (imply-ing uniform boundedness principle, i.e., resonance theorem) are the three basic principlesof the theory of functional analysis. Functional analysis greatly depends on Baire cate-gory theorem and Hahn-Banach extension theorem. But the former is a proposition of thetheory of general topology and the latter, as W. Orlicz and a scholar of our country XiaDaoxing explicitly say, belongs to pure algebra.
There is a common property among classical three basic principles of functionalanalysis, that is, treated mappings are only linear operators which are too ideal. So thethree basic principles are not applicable to nonlinear mappings. This greatly makes thetheoretical value and applications of three basic principles limited. So there have beenmore than sixty years for our hard work to generalize and improve the three basic prin-ciples. However, the three basic principles are still as before with extensions and im-provements of more than sixty years. In this dissertation, we substantially give all-aroundimprovements of the three basic principles of functional analysis. In fact, we establish thenew three basic principles for three very large families of mappings respectively. Each ofthe three families of mappings includes all linear mappings handled in the classical prin-ciples. Especially, the number of nonlinear mappings in each of the three families are notfewer than that of linear mappings. Henceforth, the three basic principles of functionalanalysis are promoted to the basic principles of pan-linear analysis with general mean-ing which make linear analysis to be a special case. The theoretical value and field ofapplications of three basic principles are promoted and expanded to a new level. The lin-ear dual theory, based on the new three basic principles, has been extended to pan-lineardual theory. Especially, the theory of pan-linear distributions has given some very newresults, and new open mapping theorem also yields some results which have very strongtheoretical value and practical significance.
Our guided thought is to try to reflect real life. For example, each of linear operatorsmakes absolutely precise dissection: f(x + tz) = f(x) + tf(z), (?)x, z∈X, t∈C,where absoluteness means“(?) x, z∈X, (?)t∈C”and precision means that the coeffi- cients 1 and t accurately appear in the dissecting result. But dissecting phenomenons inreal life are not often so absolutely precise. To re?ect this unideal real life, we definitedissecting mappings: for each x∈X, f(x + tu) = rf(x) + sf(u), where u belongsto a neighborhood of 0, t satisfies |t|≤1, and |r - 1| and |s - t| are simply controlledby |t|. Hence, the family of dissecting mappings is a very natural large extension of thefamily of linear operators. The new equicontinuity theorem is just established for thefamily of dissecting mappings. And each of linear operators has absolutely precise addi-tivity: f(x) + f(z) = f(x + z). This is very ideal too. But f(x) + f(z) = f(u) where is fairly common. Following this fact, we obtain the new openmapping theorem. The new closed graph theorem is obtained as the same idea.
New three basic principles not only greatly promote the theoretical level, but alsoexpand the field of applications of three basic principles. In the last chapter, we give someapplications of new basic principles in dual theory, distributions and the theory of Banachspaces.
引文
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110 C. Swartz. The Uniform Boundedness Principle for Abitrary Locally ConvexSpaces[J]. Proyecciones, 2007, 26(3):245～251.
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