(完整word版)泛函分析课程总结(word文档良心出品).pdf

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1、泛函分析课程总结数学与计算科学学院09 数本 5 班符翠艳2009224524 序号：26 一知识总结第七章度量空间和赋范线性空间1.度量空间的定义：设 X 是一个集合，若对于 X 中任意两个元素,x y，都有唯一确定的实数,d x y 与之相对应，而且满足1,0,0=;2,;3,d x yd x yx yd x yd y xd x yd x zd z yz、的充要条件是、对任意 都成立。则称 d 为 X 上的一个度量函数，（dX,）为度量空间，),(yxd为yx,两点间的度量。2.度量空间的例子离散的度量空间,X d设 X 是任意的非空集合，对X 中任意两点,x yX,令1,0,xyd x

2、yxy当当序列空间 S 令S表 示 实 数 列（或 复 数 列）的 全 体，对S 中 任 意 两 点12n12,.,.,.,.nxy及，令11,2 1iiiiiid x y有界函数空间 B（A）设 A 是一给定的集合，令B（A）表示 A 上有界实值（或复值）函数全体，对 B（A）中任意两点,x y，定义,()()suptAd x yx ty t可测函数空间 m(X)设 m(X)为 X 上实值（或复值）的 L 可测函数全体，m 为 L 测度，若 m X，对任意两个可测函数()()f tg t及，令()(),1()()Xf tg tdf gdtf tg t第 1 页，共 12 页,C a b 空间

3、令,C a b 表示闭区间,a b 上实值（或复值）连续函数的全体，对,C a b 中任意两点,x y，定义,max()()a t bd x yx ty t2l空间记12kkkxxxl，设2kxxl，2ykyl，定义1221,()kkkd x yyx注：度量空间中距离的定义是关键。3.度量空间中的极限，稠密集，可分空间3.1 收敛点列和极限定义：设nx是,X d 中的点列，如果存在xX,使,0limnd xxn，则称点列nx是,X d 中的收敛点列，x是点列nx的极限。注：1.度量空间,X d 中的收敛点列的极限是唯一的。2.各个度量空间中各种极限概念不完全一致（依坐标收敛，一致收敛。依测度收

4、敛等）3.2 度量空间中稠密子集和可分度量空间定义：设 X 是度量空间，E 和 M 是 X 中两个自己，令 M 表示 M 的闭包，如果 EM,那么称 M 在集 E 中稠密，当 E=X 时称 M 是 X 的一个稠密子集。如果 X 由一个可数的稠密子集，则称X 是可分空间。注：1.若 A 在 B 中稠密，B 在 C 中稠密，则 A 在 C 中稠密。2.欧氏空间 Rn、空间 Ca,b、空间pplbaL,是可分的。3.l不可分。4.完备度量空间4.1 柯西点列定义：设,XX d 是度量空间，nx是 X 中的点列，如果对任意给定的正数0，存在正整数()NN,使当 n,mN 时，必有第 2 页，共 12

5、页文档编码:CP1L2G3U9Y5 HL6J4L9A5V3 ZL3W10S2D2G9文档编码:CP1L2G3U9Y5 HL6J4L9A5V3 ZL3W10S2D2G9文档编码:CP1L2G3U9Y5 HL6J4L9A5V3 ZL3W10S2D2G9文档编码:CP1L2G3U9Y5 HL6J4L9A5V3 ZL3W10S2D2G9文档编码:CP1L2G3U9Y5 HL6J4L9A5V3 ZL3W10S2D2G9文档编码:CP1L2G3U9Y5 HL6J4L9A5V3 ZL3W10S2D2G9文档编码:CP1L2G3U9Y5 HL6J4L9A5V3 ZL3W10S2D2G9文档编码:CP1L2G3

6、U9Y5 HL6J4L9A5V3 ZL3W10S2D2G9文档编码:CP1L2G3U9Y5 HL6J4L9A5V3 ZL3W10S2D2G9文档编码:CP1L2G3U9Y5 HL6J4L9A5V3 ZL3W10S2D2G9文档编码:CP1L2G3U9Y5 HL6J4L9A5V3 ZL3W10S2D2G9文档编码:CP1L2G3U9Y5 HL6J4L9A5V3 ZL3W10S2D2G9文档编码:CP1L2G3U9Y5 HL6J4L9A5V3 ZL3W10S2D2G9文档编码:CP1L2G3U9Y5 HL6J4L9A5V3 ZL3W10S2D2G9文档编码:CP1L2G3U9Y5 HL6J4L9A

7、5V3 ZL3W10S2D2G9文档编码:CP1L2G3U9Y5 HL6J4L9A5V3 ZL3W10S2D2G9文档编码:CP1L2G3U9Y5 HL6J4L9A5V3 ZL3W10S2D2G9文档编码:CP1L2G3U9Y5 HL6J4L9A5V3 ZL3W10S2D2G9文档编码:CP1L2G3U9Y5 HL6J4L9A5V3 ZL3W10S2D2G9文档编码:CP1L2G3U9Y5 HL6J4L9A5V3 ZL3W10S2D2G9文档编码:CP1L2G3U9Y5 HL6J4L9A5V3 ZL3W10S2D2G9文档编码:CP1L2G3U9Y5 HL6J4L9A5V3 ZL3W10S2D

8、2G9文档编码:CP1L2G3U9Y5 HL6J4L9A5V3 ZL3W10S2D2G9文档编码:CP1L2G3U9Y5 HL6J4L9A5V3 ZL3W10S2D2G9文档编码:CP1L2G3U9Y5 HL6J4L9A5V3 ZL3W10S2D2G9文档编码:CP1L2G3U9Y5 HL6J4L9A5V3 ZL3W10S2D2G9文档编码:CP1L2G3U9Y5 HL6J4L9A5V3 ZL3W10S2D2G9文档编码:CP1L2G3U9Y5 HL6J4L9A5V3 ZL3W10S2D2G9文档编码:CP1L2G3U9Y5 HL6J4L9A5V3 ZL3W10S2D2G9文档编码:CP1L2

9、G3U9Y5 HL6J4L9A5V3 ZL3W10S2D2G9文档编码:CP1L2G3U9Y5 HL6J4L9A5V3 ZL3W10S2D2G9文档编码:CP1L2G3U9Y5 HL6J4L9A5V3 ZL3W10S2D2G9文档编码:CP1L2G3U9Y5 HL6J4L9A5V3 ZL3W10S2D2G9文档编码:CP1L2G3U9Y5 HL6J4L9A5V3 ZL3W10S2D2G9文档编码:CP1L2G3U9Y5 HL6J4L9A5V3 ZL3W10S2D2G9文档编码:CP1L2G3U9Y5 HL6J4L9A5V3 ZL3W10S2D2G9文档编码:CP1L2G3U9Y5 HL6J4L

10、9A5V3 ZL3W10S2D2G9文档编码:CP1L2G3U9Y5 HL6J4L9A5V3 ZL3W10S2D2G9文档编码:CP1L2G3U9Y5 HL6J4L9A5V3 ZL3W10S2D2G9文档编码:CP1L2G3U9Y5 HL6J4L9A5V3 ZL3W10S2D2G9文档编码:CP1L2G3U9Y5 HL6J4L9A5V3 ZL3W10S2D2G9文档编码:CP1L2G3U9Y5 HL6J4L9A5V3 ZL3W10S2D2G9文档编码:CP1L2G3U9Y5 HL6J4L9A5V3 ZL3W10S2D2G9文档编码:CP1L2G3U9Y5 HL6J4L9A5V3 ZL3W10S

11、2D2G9文档编码:CP1L2G3U9Y5 HL6J4L9A5V3 ZL3W10S2D2G9文档编码:CP1L2G3U9Y5 HL6J4L9A5V3 ZL3W10S2D2G9文档编码:CP1L2G3U9Y5 HL6J4L9A5V3 ZL3W10S2D2G9文档编码:CP1L2G3U9Y5 HL6J4L9A5V3 ZL3W10S2D2G9,nmd xx则称nx是 X 中的柯西点列。那么称,X d 是完备的度量空间。4.2 完备度量空间的例子l是完备度量空间 C 是完备度量空间,a bC是完备度量空间4.3 定理的证明定理：完备度量空间X 的子空间 M 是完备空间的充要条件为M 是 X 中的闭子空

12、间。证 明：设 M 是 完备 子 空 间，对 每 个 xM，存在 M 中 点 列nx，使()xx nn，由前述，nx是 M 中的柯西点列，所以在 M 中收敛，有极限的唯一性可知x M，即 MM,，所以 MM，因此 M 是 X 中的闭子空间。5.度量空间的完备化5.1 等距同构映射定义：设,X d，,X d是两个度量空间，如果存在 X 到X 上的保距映射 T，即,d Tx Tyd x y，则称,X d 和,X d等距同构，T 称为 X 到X 上的等距同构映射。5.2 度量空间的完备化定理定理：设(,)XX d是度量空间，那么一定都一定存在一个完备空间,X d，使 X 与X 的某个稠密子空间 W

13、等距同构。并且X 在等距同构的意义下时唯一的，即(,)X d也是一完备度量空间，且X 与X 的某个稠密子空间等距同构，则,X d与(,)X d等距同构。注：任一度量空间,X d 都存在唯一的完备度量空间,X d，使 X 为X 的稠密子空间。6.压缩映射第 3 页，共 12 页文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2

14、S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10

15、S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H

16、3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9

17、T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P

18、10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X

19、8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6

20、G9T2S96.1 压缩映射定义：设 X 是度量空间，T 是 X 到 X 中的映射，如果存在一个数，01，使得对所有的,x yX，,d Tx Tyd x y（1）则称 T 是压缩映射6.2 压缩映射定理定理：设 X 是完备的度量空间，T 是 X 上的压缩映射，那么T 有且只有一个不动点（就是说，方程 Txx，有且只有一个解）。证明：设0 x是 X 中任意一点，令10 xTx，221010,.,.nnnxTxT xxTxT x。我们证明点列nx是 X 中柯西点列，事实上，111,(,)mmmmmmd xxd Tx Txd xx21212(,)(,)mmmmd TxTxd xx（2）10.(,)m

21、d x x由三点不等式，当nm 时，1121(,)(,)(,).(,)mnmmmmnnd xxd xxd xxd xx1101(.)(,)mmnd xx011(,).1n mmd xx?因 01，所以11n m，于是得到01(,)(,)1mmnd xxd xx（nm）(3)所以当,mn时，(,)0mnd xx，即nx是 X 中柯西点列，由X 完备，存在 xX，使()mxx m，又由三点不等式和条件（1），我们有1(,)(,)(,)(,)(,).mmmmd x Txd x xd xTxd x xd xx上面不等式右端当m时趋于 0，所以(,)0,d x Tx即Txx第 4 页，共 12 页文档编

22、码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O

23、3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2

24、 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文

25、档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P

26、5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9

27、S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S

28、9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9下证唯一性。如果又有,xX使T xx,则由条件（1），(,)(,)(,).d x xd Tx T xd x x因1，所以必有(,)0d x x，即xx。注：1.X 是完备的度量空间2.T 是压缩映射3.压缩定理可以推导出隐函数存在定理4.压缩映射原理可以证明常微分方程解

29、得存在性和唯一性定理7.赋范线性空间和巴拿赫空间7.1 赋范线性空间定义：设 X 是实（或复）的线性空间，如果对每个向量xX，有一个确定的实数，记为x 与之对应，并满足1.0,00;2.,3.,.xxxxxxyxyx yX且等价于其中为任意实（复）数；则称 x 为向量x的范数，称 X 按范数 x 成为赋范线性空间。设nx是 X 中点列，如果存在 xX，使0()nxxn，则称nx依范数收敛于x，记为()limnnnxx nxx或。如果令(,)d x yxy(,)x yX即nx依范数收敛于x等价于nx按距离(,)d x y收敛于x，称(,)d x y为由范数x 导出的距离。注：完备的赋范线性空间称

30、为巴拿赫空间7.2 几种常见的巴拿赫空间欧式空间nR对每一个12,.,nnxR，定义范数222.12nx（1）又因nR 完备，x 是nR 中范数。故nR 按（1）式中范数成为巴拿赫空间。空间,C a b第 5 页，共 12 页文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8

31、H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G

32、9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5

33、P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7

34、X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V

35、6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:C

36、B5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9对每一个,C a bx，定义max()

37、a t bxx t（2）,C a b 按（2）式中的范数成为巴拿赫空间。空间l对每一个12,.,xl，定义supjjx（3）l按（3）式中的范数成为巴拿赫空间。空间,pLa b(1)p对于每个,pfLa b，定义1()bpppaff tdt（4）,pLa b(1)p按（4）式中的范数成为巴拿赫空间。空间pl对每一个12,.,plx，定义111pppix（5）pl按（5）式中的范数成为巴拿赫空间。7.3 两个重要的不等式和两条定理(1)霍尔德不等式设111,1,ppgpqpfLa bLa b，那么()()f t g t在,a b上L 可积，并且()()bpqaf t g t dtfg（2）闵可夫

38、斯基不等式设1p，,pf gLa b，那么,pfgLa b，并且成立不等式第 6 页，共 12 页文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:

39、CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3

40、HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 Z

41、F6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编

42、码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O

43、3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2

44、 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9()()pppf tg tfg定理 1：当1p时，,pLa b按（4）式中范数pf成为赋范线性空间。定理 2：,pLa b(1)p是巴拿赫空间7.4 有限维赋范线性空

45、间的性质定理 3：设X是 n 维赋范线性空间，12,.,ne ee是X的一组基，则存在常数M 和M，使得对一切1kkkxe，有1221()nkkMxMx推论 1：设在有限维线性空间上定义了两个范数x 和1x，那么必存在常数M和M，使得1M xxxM拓扑同构的定义：设11,Rx和22,Rx是两个赋范线性空间。如果存在从1R到2R上的线性映射和正数1c，2c，使得对一切1xR，有21212xxxcc则称11,Rx和22,Rx是两个赋范线性空间是拓扑同构推论 2：任何有限维赋范空间都和同维数欧式空间拓扑同构，相同维数的有限维赋范空间彼此拓扑同构。8.度量空间、赋范线性空间、巴拿赫空间的区别与联系赋范

46、线性空间一定是度量空间，反之不一定成立。度量空间按照加法和数乘运算成为线性空间，而且度量空间中的距离如果是由范数导出的，那么这个度量空间就是赋范线性空间。赋范线性空间与巴拿赫空间的联系与区别：完备的赋范线性空间是巴拿赫空间。巴拿赫空间一定是赋范线性空间，反之不一定成立。巴拿赫空间一定是度量空间，反之不一定成立。巴拿赫空间满足度量空间的所有性质。巴拿赫空间由范数导出距离，而且满足加法和数乘的封闭性。满足完备性，则要求每个柯西点列都在空间中收敛。度量空间中距离要满足三个性质：非负线性、对称性、三点不等式，因此距离),(yxd的定义是重点。赋范线性空间中范数要满足：非负性、线性性、三角不第 7 页，

47、共 12 页文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB

48、5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG

49、7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6

50、V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码:CB5P10S1P5O3 HG7X8H3M9S2 ZF6V6G9T2S9文档编码: