泛函分析讲稿new

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1、泛函分析讲稿2011年7月11日目录第第第一一一章章章度度度量量量空空空间间间11.1度量空间.....................................11.1.1定义....................................11.1.2子空间...................................71.1.3积空间...................................71.1.4极限..................................

2、..81.2赋范线性空间、内积空间............................101.2.1线性空间.................................101.2.2赋范线性空间...............................121.2.3内积空间.................................131.3度量空间中的点集................................161.3.1内点、开集........................

3、........161.3.2极限点、闭集...............................201.3.3点集间的距离...............................221.3.4连续映照.................................221.3.5连续曲线.................................241.4完备性.......................................261.4.1完备性的概念..............

4、.................261.4.2闭球套定理................................301.4.3Baire定理.................................311.5不动点定理....................................341.5.1压缩映照原理...............................341.5.2应用....................................371.6稠密性.......

5、................................411.6.1稠密性...................................41iii目录1.6.2可分空间.................................441.7紧性........................................451.7.1相对列紧集................................451.7.2完全有界集.............................

6、...461.7.3Arzela-Ascoli定理............................501.7.4列紧集...................................531.7.5紧集上的连续映照............................551.8习题........................................58第第第二二二章章章线线线性性性泛泛泛函函函，，，Hahn-Banach定定定理理理612.1线性泛函..................

7、...................612.1.1定义....................................612.1.2线性泛函的几何意义...........................622.2Hahn-Banach定理................................632.2.1凸集....................................632.2.2半范数和Minkowski泛函.........................642.2.3Ha

8、hn-Banach定理............................662.2.4凸集的可分离性.............................692.2.5线性泛函的连续性和有界性.......................712.2.6赋范空间中的Hahn-Banach定理.....................752.2.7赋范线性空间中的分离性定理...