II T^u/I^ ^r Kb,u)f dyiĂb) Cl-8)
Bx(^)= ye (co) b
^-^ n n l a dong l u f t v o i Ạ
Chỷng minh vl 1 nhỷng diijợc vao mpt khong g i a n L ( S , X Í M ) v ụ i
P P (B* 2 t / ^ ) l a mpt khong g i a n do nao do t a thõy t o n t f i mpt dõy (e )
t r o n g L (êÍ 7" ,./u) va mpt d a y (g ) t r o n g L ( b , 2 ÍÂ) cỷng v ụ i P "^ '' n p ' mpt õnh :xf I : \J^'\ — ^ ^ ợ^Í 2^ \f^ ) saợ> cho Ih = 21(I^Íg )'^ va rv=.( n n | ( I h | t ^ = [d(h)]^ = 2 ; i ( i h , g ) i ^ Ca-6) D|Lt T r I T^ : X ẻ2) Ý ~ > L ( S , Z Íyụ ) • Tẻr ( i - 2 ) va (a-6.), t a thu dý^c E e x p { i T ^ t t j = e x p J - l|Tụ|p'l = e x p ợ - Y^ |{T^tÍ6j,| ^ j (2-7>.- Nhý dõ chạ r a t r o n g chỷng minh djnh 1 y 2Ậ.4 co t o n t f i mpt õn% Xf t u y ờ n t ợ n h l i e n t y e G : X —ợ> oL(*^r^) Í^ dụ M l õ dp do ngõu n h i ờ n p-on djnh doi xỷng t r e n ( S , 5L * M ) s^° *^^*^ ^^^ °*P^ y ờ ^ ' T ( x < f > y ) ( t ) z: / ' G x ( t ) Í y ) vSi y ^ - h õ u h ờ t t . v ụ i moi g cụ djnh thupc L CSÍ X Íyu ) . t a x ờ t õzih xf t u y ờ n t ợ n h Vg
tiợ X vao Y i dýjic cho nhii sau
Vg(x) = (g(t)Gx(t)dyụ(t) ẻ2-81 Tich phan Bociiner (2-8) ton t ^ i yl |IQx(t).|l^ d ^ ( t ) < oọTẻi
(2-8) ta cụ
llVg(x)|l ^ llgll â rilGx(t)llPdyuL(t)j ^^P ợ Cll ei;|IGx:|l^ <C CIglIlIGlIflxll Dieu nay chõng to Tg ^LẻX^Y) - ^ ll Vg || .$ C II g ll II G ll -S^t b =
r V g ^ . T a c ụ (h^x^y) - (Vg^(x)Íy) =. ("(g (aGx(tKyMyH(t) -
- yT(x Ậ ) y ) ( t ) g ^ ( t ) dyu(t) = ( T ( X CS) y)„g ") . ợfiy t h i tợợ (2-7) (i|t u = X Ẽ y ta cụ ợfiy t h i tợợ (2-7) (i|t u = X Ẽ y ta cụ
C30 oo
E exp[i(AxÍy)j = exp \-'^\(r(x Ẽ y),Bj\ ""] = e x p ợ - ^ |(b^x,y)l ^ j
(2-9)
Ta chỷng minh khang djnh l).Eam dac trýng cua tong rieng Y^ Ẫ b x
4 4 7*1 Y\
ham lõ expợ- 2 2 K^ x,y):|* ( .Tẻi (2-9) ta suy ra khi m —Í oc ha
y 'yyy
dfc trýng cua tong riờng Y Ẫ ^ x hpi ty tụi hõm dfc trýng cua Ax-
Theo djnh ly Ito-Iợisio ta co chuoi \ Q h x hpi ty h.c.c. trong Ỵ n— /
Ta chỷng minh khõng djnh 2).Vểi (x ^'^d XÍ (y, )^C YÍ l õ dõy hýu tifn k ' k 1
ta cụ
Ậo
E expji Ị t ^ ( B x ^ . y ^ ) ] - E expji ê ê , ^ ^ V k ^ V ^ i ] "
— E expiiT u"l = E expJi Ị i t (Ax ,y )( •Vly A va B dong lu^t . L A J ^ f c s i k k k - - ^
THlf MịC
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Phy l y c riay g i ợ i thí^u ^.-^t so khai ni^!:n,djnri i y cua l y t h u y e t xac
s u õ t t r e n khong gian Banaeh dýp'c s Ị dyng t r o n g lugui an.CỊc chỷng minh cung nhý cae thong t i n l i e n quan,ngýcJi dpe co the tim dpe t r o n g [ I J
[5], [28] , [60}, I5l]
t Z^ ^
I . B i e n n^õu n h i ờ n ^ i õ t r j t r e n khon^ põan Banaeh
Giõ s i CXI , ?" ) l õ liipt khong g i a n do dýp'c ,P l õ mpt dp de xac s u õ t t r e n (-TI , . ? ) . Ta gpi bp ba ( ợ ^ , 5^ ,P) l õ mpt khong gian xõc s u õ t . C h o E l a mpt khong gian Banaeh t r a n g bJ S ' - d f i so <-'-> eac t f p
B o r e l . ạr.h xf f : S^ —> B dýp'c gpi l õ mpt b i f u ngau n h i ờ n B- g i õ t r J nờu f l õ do dý^z{ t a c l õ f'^B) ê ? vưi mpi B e ^ ) võ
f cụ mien g i õ t r J l õ mpt t f p kha l y . T f p hp'p cae b i e n ngõu n h i ờ n B - g i a t r J dýpc ky hipu l õ ! ' ( / ! ) , L ^ ( - n ) l a iipt iihong g i a n v e c t c tepe (khong l o i dJa phýdng) m e t r i e hõ dýp*c bụi khoang cach Proi-ýiorov : Day
( '^ , ^ 0 J , ờ
f cõc b i e n ngõu n h i ờ n E - g i a t r J hpi t y t d i f neu vdi mpi t > O t a eo