講者：許永昌 老師 1

Contents Completeness Bessel’s inequality Schwarz Inequality Summary 2

Completeness ( 請昱讀 P649~P651) 3 這裡內積不用 Dirac braket 是因為 已經定義 =  (r  r’) 了。 若通用，邏輯上會發生錯亂。 這裡內積不用 Dirac braket 是因為 已經定義 =  (r  r’) 了。 若通用，邏輯上會發生錯亂。

Completeness & Hilbert Space ( 請預讀 P650 & P658) 4 ~~~~~~~~~ ( 這很重要 ) If  (x)=1, we can write If  (x)=1, we can write

Completeness ( 請預讀 P883~P884) 5 在此，內積並不考慮 weight ， 原因是 H  m = m  m 。

Bessel’s Inequality ( 請預讀 P651~P652) 6

Schwarz Inequality ( 請預讀 P652~P654) 7

Summary --- Vector space, Completeness ( 請預讀 P654~P658) (finite) Vector SpaceHilbert Space Basis: {e i } span the linear vector space {  i } span the linear vector (function) space Vector space: {1.addition, 2.multiplication of a scalar (a, b, 0, 1, -1)}  {1.commutative, 2.associative, 3.null vector and negative vector, 4. distributive} 1,1: f+g=g+f, 1,2: f+(g+h)=(f+g)+h, 1,3: f+0=f 12,4: a(f+g)=af+ag & (a+b)f=af+bf 2,1: af=fa 2,2: (ab)f=a(bf) 2,3: 0*f=0, 1*f=f, -1*f=-f. Complete: Inner product: c  d=  c i *d i. (f,g+ah)=(f,g)+a(f,h), (f,g)=(g,f)*. Orthogonality: e i  e j =0 if i  j  i *  j  dx=0 if i  j 8

Summary --- Vector space, Completeness (continue) (finite) Vector SpaceHilbert Space Norm: |c|=sqrt(c  c) ||f||=(f,f) ½ Bessel’s inequality: c  c   |c i | 2. (f,f)  |(f,  i )| 2. Schwarz’ inequality: |c  d|  |c|  |d| |(f,g)| 2  ||f||  ||g|| Infinite-dimensional linear vector space l 2 : c i =c  e i. Required:  |c i | 2 < . Infinite-dimensional linear vector (function) space L 2 : f i =(  i,f). f=  i (  i,f) : (almost everywhere) 9

Homework (9.4.1e) (9.4.3e) (9.4.9e) (9.4.10e) 10

Nouns 11