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chpp18 - 19 l C o y 3 n A K K k — l C 3 l C K X ~ ^ 1 A U...

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Unformatted text preview: 19 l C { o y 3 , ? n A ; . K , K k { — l C { . 3 l C { , K X ~ ^ : 1. A U , K U m . K U , | m ? 5 . 2. U X . 5 l n A K , l z l C { @ . 19.1 S m m 5 m x y 8 V x ∈ V y ∈ V e u ? ( E ) α , β α x + β y ∈ V K V ( E ) m 5 m , m . Example 19.1 ( n m ) V 3 = { x i + y j + z k } Example 19.2 C t k E X 8 . Example 19.3 (Laplace ) ∇ 2 u = 0 S m e 3 ( E K ) m I ( x , y ) . v : 1. ( x , y ) = ( y , x ) * 2. ( α x + β y , z ) = α * ( x , z ) + β * ( y , z ) α β K I ; 3. u ? x ( x , x ) ≥ = x = , ( x , x ) = 0 K ( x , y ) x , y S . S m S m . k S m ( A p ) m (Euclidean space); k S E m j m (unitary space). Example 19.4 ( n m ) ( r 1 , r 2 ) = x 1 x 2 + y 1 y 2 + z 1 z 2 Example 19.5 C t k E X m . e ≤ t ≤ 1 , K d m ( ) x ( t ) y ( t ) S ( x ( t ) , y ( t )) = Z 1 x * ( t ) y ( t ) ρ ( t ) d t, 3 ≤ t ≤ 1 S , ρ ( t ) ≥ , ρ 6≡ e ρ ( t ) ≡ 1 ( x ( t ) , y ( t )) = Z 1 x * ( t ) y ( t ) d t. 1 S 1 1 ^ , w , m j m , g S o . 3 d : , k ^ 3. r ( x , x ) 1 / 2 = k x k (1) x ( = x “ ”). 3 S , V g : = ( x , y ) = 0 , x , y . 8 e u k i j , ( x i , x j ) = δ ij K | { x 1 , x 2 , ···} 8 . 3 n m V k | 8 { x 1 , x 2 , ··· , x k } , k ≤ n. K k k X i =1 α i x i k f m . K u ? x ∈ V , α i = ( x i , x ), K ∑ k i =1 α i x i x 3 k f m K . w , A k x- k X i =1 α i x i 2 ≡ x- k X i =1 α i x i , x- k X i =1 α i x i = ( x , x )- k X i =1 α * i α i- k X i =1 α i α * i + k X i,j =1 α * i α j δ ij = ( x , x )- k X i =1 α * i α i ≥ , d d — l ( x , x ) ≥ k X i =1 | ( x i , x ) | 2 (2) = k x k 2 ≥ k X i =1 | ( x i , x ) | 2 . (3) 8 5 ' . , ? | 5 ' 8 z . n m ? | n 8 d m , 8 , ( 5 I O ) 3 k m , 8 8 7 , m . 3 S $ ^ , e k m , | 8 , ^ l 5 O : l , = u ? x ∈ V , k k x k 2 = k X i =1 | ( x i , x ) | 2 . l (Parseval) , 8 | 7 ^ . 2 19.2 m m a A m : m . ~ X , 3 m ( , 4 m a ≤ x ≤ b ) E f ( x ) 8 . Z b a | f ( x ) | 2 d x < ∞ y 5 | E, . d , 8 m . ~ X , E , | f 1 ( x ) + f 2 ( x ) | 2 + | f 1 ( x )- f 2 ( x ) | 2 = 2 | f 1 ( x ) | 2 + | f 2 ( x ) | 2 , | f 1 ( x ) + f 2 ( x ) | 2 ≤ 2 | f 1 ( x ) | 2 + | f 2 ( x ) | 2 . Theorem 19.1. f 1 ( x ) f 2 ( x ) ( ) m , K Z b a f * 1 ( x ) f 2 ( x )d x 3 ....
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