ÿþP r z e s t r z e D u n o r m o w a n a
A n a l i z a f u n k c j o n a l n a - w y k Ba d 2
Z o f i a L e w a n d o w s k a
I M a t e m a t y k a S D S
s p e c j a l i z a c j a n a u c z y c i e l s k a
Z o f i a L e w a n d o w s k a A n a l i z a f u n k c j o n a l n a - w y k Ba d 2
P r z e s t r z e D u n o r m o w a n a
S p i s t r e [c i
1
P r z e s t r z e D u n o r m o w a n a
D e f i n i c j e
W Ba s n o [c i
P r z y k Ba d y
N i e r ó w n o [c i p o m o c n i c z e
N o r m a w p r z e s t r z e n i l p
p
P r z e s t r z e D l n
P r z e s t r z e D B a n a c h a
Z o f i a L e w a n d o w s k a A n a l i z a f u n k c j o n a l n a - w y k Ba d 2
D e f i n i c j e
W Ba s n o [c i
P r z y k Ba d y
P r z e s t r z e D u n o r m o w a n a N i e r ó w n o [c i p o m o c n i c z e
N o r m a w p r z e s t r z e n i l p
P r z e s t r z e D l p
n
P r z e s t r z e D B a n a c h a
P r z e s t r z e D u n o r m o w a n a
D e f i n i c j a n o r m y
N i e c h X b d z i e p r z e s t r z e n i l i n i o w n a d c i a Be m l i c z b r z e c z y w i s t y c h
l u b z e s p o l o n y c h . F u n k c j · : X ’! [ 0 , ") n a z y w a m y n o r m ,
j e |e l i s p e Bn i a n a s t p u j c e w a r u n k i :
( N 1 ) x = 0 Ð!Ò! x = ˜,
( N 2 ) »x = | » | · x ,
( N 3 ) x + y x + y .
Z o f i a L e w a n d o w s k a A n a l i z a f u n k c j o n a l n a - w y k Ba d 2
D e f i n i c j e
W Ba s n o [c i
P r z y k Ba d y
P r z e s t r z e D u n o r m o w a n a N i e r ó w n o [c i p o m o c n i c z e
N o r m a w p r z e s t r z e n i l p
P r z e s t r z e D l p
n
P r z e s t r z e D B a n a c h a
P r z e s t r z e D u n o r m o w a n a
D e f i n i c j a n o r m y
N i e c h X b d z i e p r z e s t r z e n i l i n i o w n a d c i a Be m l i c z b r z e c z y w i s t y c h
l u b z e s p o l o n y c h . F u n k c j · : X ’! [ 0 , ") n a z y w a m y n o r m ,
j e |e l i s p e Bn i a n a s t p u j c e w a r u n k i :
( N 1 ) x = 0 Ð!Ò! x = ˜,
( N 2 ) »x = | » | · x ,
( N 3 ) x + y x + y .
D e f i n i c j a p r z e s t r z e n i u n o r m o w a n e j
P a r ( X , · ) n a z y w a m y p r z e s t r z e n i u n o r m o w a n .
Z o f i a L e w a n d o w s k a A n a l i z a f u n k c j o n a l n a - w y k Ba d 2
D e f i n i c j e
W Ba s n o [c i
P r z y k Ba d y
P r z e s t r z e D u n o r m o w a n a N i e r ó w n o [c i p o m o c n i c z e
N o r m a w p r z e s t r z e n i l p
P r z e s t r z e D l p
n
P r z e s t r z e D B a n a c h a
W Ba s n o [c i n o r m y
1
| x - y | x - y
m m
2
x n x n
n = 1 n = 1
Z o f i a L e w a n d o w s k a A n a l i z a f u n k c j o n a l n a - w y k Ba d 2
D e f i n i c j e
W Ba s n o [c i
P r z y k Ba d y
P r z e s t r z e D u n o r m o w a n a N i e r ó w n o [c i p o m o c n i c z e
N o r m a w p r z e s t r z e n i l p
P r z e s t r z e D l p
n
P r z e s t r z e D B a n a c h a
W Ba s n o [c i n o r m y
1
| x - y | x - y
m m
2
x n x n
n = 1 n = 1
Z o f i a L e w a n d o w s k a A n a l i z a f u n k c j o n a l n a - w y k Ba d 2
D e f i n i c j e
W Ba s n o [c i
P r z y k Ba d y
P r z e s t r z e D u n o r m o w a n a N i e r ó w n o [c i p o m o c n i c z e
N o r m a w p r z e s t r z e n i l p
P r z e s t r z e D l p
n
P r z e s t r z e D B a n a c h a
W Ba s n o [c i n o r m y
T w i e r d z e n i e
K a |d a p r z e s t r z e D u n o r m o w a n a X j e s t p r z e s t r z e n i m e t r y c z n
z m e t r y k z d e f i n i o w a n w z o r e m :
( x , y ) = x - y ,
g d z i e x " X , y " X .
Z o f i a L e w a n d o w s k a A n a l i z a f u n k c j o n a l n a - w y k Ba d 2
D e f i n i c j e
W Ba s n o [c i
P r z y k Ba d y
P r z e s t r z e D u n o r m o w a n a N i e r ó w n o [c i p o m o c n i c z e
N o r m a w p r z e s t r z e n i l p
P r z e s t r z e D l p
n
P r z e s t r z e D B a n a c h a
W Ba s n o [c i n o r m y
T w i e r d z e n i e
K a |d a p r z e s t r z e D u n o r m o w a n a X j e s t p r z e s t r z e n i m e t r y c z n
z m e t r y k z d e f i n i o w a n w z o r e m :
( x , y ) = x - y ,
g d z i e x " X , y " X .
U w a g a
W s z y s t k i e p o j c i a t o p o l o g i c z n e z d e f i n i o w a n e w p r z e s t r z e n i a c h
m e t r y c z n y c h o d n o s z s i r ó w n i e | d o p r z e s t r z e n i u n o r m o w a n y c h .
Z o f i a L e w a n d o w s k a A n a l i z a f u n k c j o n a l n a - w y k Ba d 2
D e f i n i c j e
W Ba s n o [c i
P r z y k Ba d y
P r z e s t r z e D u n o r m o w a n a N i e r ó w n o [c i p o m o c n i c z e
N o r m a w p r z e s t r z e n i l p
P r z e s t r z e D l p
n
P r z e s t r z e D B a n a c h a
P r z y k Ba d y
X p r z e s t r z e D l i n i o w a m - w y m i a r o w a z b a z
B = { e 1 , e 2 , . . . , e m } ,
m
x = | ±n | ,
n = 1
g d z i e x " X m a j e d n o z n a c z n e p r z e d s t a w i e n i e
x = ±1 e 1 + ±2 e 2 + . . . + ±m e m
Z o f i a L e w a n d o w s k a A n a l i z a f u n k c j o n a l n a - w y k Ba d 2
D e f i n i c j e
W Ba s n o [c i
P r z y k Ba d y
P r z e s t r z e D u n o r m o w a n a N i e r ó w n o [c i p o m o c n i c z e
N o r m a w p r z e s t r z e n i l p
P r z e s t r z e D l p
n
P r z e s t r z e D B a n a c h a
P r z y k Ba d y
m , x = ( t n ) n "N " m
x = s u p | t n | ,
n "N
c l u b c 0 , x = ( t n ) n "N
x = s u p | t n | ,
n "N
Z o f i a L e w a n d o w s k a A n a l i z a f u n k c j o n a l n a - w y k Ba d 2
D e f i n i c j e
W Ba s n o [c i
P r z y k Ba d y
P r z e s t r z e D u n o r m o w a n a N i e r ó w n o [c i p o m o c n i c z e
N o r m a w p r z e s t r z e n i l p
P r z e s t r z e D l p
n
P r z e s t r z e D B a n a c h a
P r z y k Ba d y
m , x = ( t n ) n "N " m
x = s u p | t n | ,
n "N
c l u b c 0 , x = ( t n ) n "N
x = s u p | t n | ,
n "N
Z o f i a L e w a n d o w s k a A n a l i z a f u n k c j o n a l n a - w y k Ba d 2
D e f i n i c j e
W Ba s n o [c i
P r z y k Ba d y
P r z e s t r z e D u n o r m o w a n a N i e r ó w n o [c i p o m o c n i c z e
N o r m a w p r z e s t r z e n i l p
P r z e s t r z e D l p
n
P r z e s t r z e D B a n a c h a
P r z y k Ba d y
C [ a , b ] , f " C [ a , b ]
f = s u p | f ( x ) | ,
x "[ a , b ]
l 1 , x = ( t n ) n "N " l 1
"
x = | t n | ,
n = 1
l p , g d z i e p > 1 , x = ( t n ) n "N " l p
1
"
p
x = | t n | p ,
n = 1
Z o f i a L e w a n d o w s k a A n a l i z a f u n k c j o n a l n a - w y k Ba d 2
D e f i n i c j e
W Ba s n o [c i
P r z y k Ba d y
P r z e s t r z e D u n o r m o w a n a N i e r ó w n o [c i p o m o c n i c z e
N o r m a w p r z e s t r z e n i l p
P r z e s t r z e D l p
n
P r z e s t r z e D B a n a c h a
P r z y k Ba d y
C [ a , b ] , f " C [ a , b ]
f = s u p | f ( x ) | ,
x "[ a , b ]
l 1 , x = ( t n ) n "N " l 1
"
x = | t n | ,
n = 1
l p , g d z i e p > 1 , x = ( t n ) n "N " l p
1
"
p
x = | t n | p ,
n = 1
Z o f i a L e w a n d o w s k a A n a l i z a f u n k c j o n a l n a - w y k Ba d 2
D e f i n i c j e
W Ba s n o [c i
P r z y k Ba d y
P r z e s t r z e D u n o r m o w a n a N i e r ó w n o [c i p o m o c n i c z e
N o r m a w p r z e s t r z e n i l p
P r z e s t r z e D l p
n
P r z e s t r z e D B a n a c h a
P r z y k Ba d y
C [ a , b ] , f " C [ a , b ]
f = s u p | f ( x ) | ,
x "[ a , b ]
l 1 , x = ( t n ) n "N " l 1
"
x = | t n | ,
n = 1
l p , g d z i e p > 1 , x = ( t n ) n "N " l p
1
"
p
x = | t n | p ,
n = 1
Z o f i a L e w a n d o w s k a A n a l i z a f u n k c j o n a l n a - w y k Ba d 2
D e f i n i c j e
W Ba s n o [c i
P r z y k Ba d y
P r z e s t r z e D u n o r m o w a n a N i e r ó w n o [c i p o m o c n i c z e
N o r m a w p r z e s t r z e n i l p
P r z e s t r z e D l p
n
P r z e s t r z e D B a n a c h a
N i e r ó w n o [c i p o m o c n i c z e
| a + b | p 2 p ( | a | p + | b | p ) , p > 1 , a , b " R
1 1 1 1
x y x p + y q , x 0 , y 0 , p > 1 , q > 1 , + = 1
p q p q
n i e r ó w n o [ H ö l d e r a d l a s z e r e g ó w
1
1
" " "
p q
| t n Än | | t n | p · | Än | q
n = 1 n = 1 n = 1
1 1
x = ( t n ) n "N " l p , y = ( Än ) n "N " l q , p > 1 , q > 1 , + = 1
p q
Z o f i a L e w a n d o w s k a A n a l i z a f u n k c j o n a l n a - w y k Ba d 2
D e f i n i c j e
W Ba s n o [c i
P r z y k Ba d y
P r z e s t r z e D u n o r m o w a n a N i e r ó w n o [c i p o m o c n i c z e
N o r m a w p r z e s t r z e n i l p
P r z e s t r z e D l p
n
P r z e s t r z e D B a n a c h a
N i e r ó w n o [c i p o m o c n i c z e
| a + b | p 2 p ( | a | p + | b | p ) , p > 1 , a , b " R
1 1 1 1
x y x p + y q , x 0 , y 0 , p > 1 , q > 1 , + = 1
p q p q
n i e r ó w n o [ H ö l d e r a d l a s z e r e g ó w
1
1
" " "
p q
| t n Än | | t n | p · | Än | q
n = 1 n = 1 n = 1
1 1
x = ( t n ) n "N " l p , y = ( Än ) n "N " l q , p > 1 , q > 1 , + = 1
p q
Z o f i a L e w a n d o w s k a A n a l i z a f u n k c j o n a l n a - w y k Ba d 2
D e f i n i c j e
W Ba s n o [c i
P r z y k Ba d y
P r z e s t r z e D u n o r m o w a n a N i e r ó w n o [c i p o m o c n i c z e
N o r m a w p r z e s t r z e n i l p
P r z e s t r z e D l p
n
P r z e s t r z e D B a n a c h a
N i e r ó w n o [c i p o m o c n i c z e
| a + b | p 2 p ( | a | p + | b | p ) , p > 1 , a , b " R
1 1 1 1
x y x p + y q , x 0 , y 0 , p > 1 , q > 1 , + = 1
p q p q
n i e r ó w n o [ H ö l d e r a d l a s z e r e g ó w
1
1
" " "
p q
| t n Än | | t n | p · | Än | q
n = 1 n = 1 n = 1
1 1
x = ( t n ) n "N " l p , y = ( Än ) n "N " l q , p > 1 , q > 1 , + = 1
p q
Z o f i a L e w a n d o w s k a A n a l i z a f u n k c j o n a l n a - w y k Ba d 2
D e f i n i c j e
W Ba s n o [c i
P r z y k Ba d y
P r z e s t r z e D u n o r m o w a n a N i e r ó w n o [c i p o m o c n i c z e
N o r m a w p r z e s t r z e n i l p
P r z e s t r z e D l p
n
P r z e s t r z e D B a n a c h a
N i e r ó w n o [c i p o m o c n i c z e
x = ( t n ) n "N " l p = Ò! y = ( | t n | p - 1 ) n "N " l q
1 1
p > 1 , q > 1 , + = 1
p q
n i e r ó w n o [ M i n k o w s k i e g o d l a s z e r e g ó w
1 1
1
" " "
p p p
| t n + Än | p | t n | p + | Än | p
n = 1 n = 1 n = 1
x = ( t n ) n "N " l p , y = ( Än ) n "N " l p , p > 1 ,
Z o f i a L e w a n d o w s k a A n a l i z a f u n k c j o n a l n a - w y k Ba d 2
D e f i n i c j e
W Ba s n o [c i
P r z y k Ba d y
P r z e s t r z e D u n o r m o w a n a N i e r ó w n o [c i p o m o c n i c z e
N o r m a w p r z e s t r z e n i l p
P r z e s t r z e D l p
n
P r z e s t r z e D B a n a c h a
N i e r ó w n o [c i p o m o c n i c z e
x = ( t n ) n "N " l p = Ò! y = ( | t n | p - 1 ) n "N " l q
1 1
p > 1 , q > 1 , + = 1
p q
n i e r ó w n o [ M i n k o w s k i e g o d l a s z e r e g ó w
1 1
1
" " "
p p p
| t n + Än | p | t n | p + | Än | p
n = 1 n = 1 n = 1
x = ( t n ) n "N " l p , y = ( Än ) n "N " l p , p > 1 ,
Z o f i a L e w a n d o w s k a A n a l i z a f u n k c j o n a l n a - w y k Ba d 2
D e f i n i c j e
W Ba s n o [c i
P r z y k Ba d y
P r z e s t r z e D u n o r m o w a n a N i e r ó w n o [c i p o m o c n i c z e
N o r m a w p r z e s t r z e n i l p
P r z e s t r z e D l p
n
P r z e s t r z e D B a n a c h a
P r z e s t r z e D l p
N o r m a w p r z e s t r z e n i l p
1
"
p
x = | t n | p
n = 1
g d z i e x = ( t n ) n "N " l p
Z o f i a L e w a n d o w s k a A n a l i z a f u n k c j o n a l n a - w y k Ba d 2
D e f i n i c j e
W Ba s n o [c i
P r z y k Ba d y
P r z e s t r z e D u n o r m o w a n a N i e r ó w n o [c i p o m o c n i c z e
N o r m a w p r z e s t r z e n i l p
P r z e s t r z e D l p
n
P r z e s t r z e D B a n a c h a
P r z e s t r z e D l p
N o r m a w p r z e s t r z e n i l p
1
"
p
x = | t n | p
n = 1
g d z i e x = ( t n ) n "N " l p
U w a g a
m 0 ‚" l 1 ‚" l p ‚" l p ‚" c 0 ‚" c ‚" m ‚" s
g d z i e 1 <