real-analysis-class44 实变函数论应用数学讲义.pdf

實變函數論─應用數學系 吳培元老師    Class 44 Chap.4. Banach spaces Functional analysis: Consider spaces of functions. topology + algebra (1) space operator: duality theory Reason: In Cn , inner product    In Hilbert space, ,        In Banach space B,    But (B,B*), x , f f x      (2) operator: spectral theory (compact, normal) spectrum-eigenvalue. T I x y or Tx y    real vector space : +, . over R  X  complex vector space : +, . over C  Let F R or C independence, span, basis, dimension, normed space:     x x : X R    1 x 0 x ,   : 2 x 0 x 0,    3 x . x ,       4 x y x y    x, y x y metric   Def. Banach space: X ,  complete   Def. X metric linear space if (1) X vector space, (2) X metric space with  , x y  x y X X X (3) ,   x x F X X ,   are conti. from      R or C Def. X Frechet space if (1) X metric linear space; (2)  x , y  x z, y z , x, y, z X     (3) X com