Nerivnist Geldera v funkcionalnomu analizi i sumizhnih disciplinah ce fundamentalna vlastivist prostoriv L p displaystyle L p Zmist 1 Formulyuvannya 2 Dovedennya 2 1 Lema 2 2 Vlasne dovedennya 3 Chastkovi vipadki 3 1 Nerivnist Koshi Bunyakovskogo 3 2 Evklidiv prostir 3 3 Prostir lp 3 4 Jmovirnisnij prostir 4 Div takozh 5 DzherelaFormulyuvannya RedaguvatiNehaj X F m displaystyle X mathcal F mu nbsp prostir z miroyu L p L p X F m displaystyle L p equiv L p X mathcal F mu nbsp prostir funkcij viglyadu f X R displaystyle f X to mathbb R nbsp iz skinchennim integrovnim p displaystyle p nbsp im stepenem Todi v ostannomu viznachena norma f p X f x p m d x 1 p p 1 displaystyle f p left int limits X f x p mu dx right 1 p qquad p geq 1 nbsp Nehaj f L p g L q p q 1 1 p 1 q 1 displaystyle f in L p quad g in L q quad p q geq 1 quad frac 1 p frac 1 q 1 nbsp Todi f g L 1 f g 1 f p g q displaystyle f cdot g in L 1 quad f cdot g 1 leq f p cdot g q nbsp Dovedennya RedaguvatiLema Redaguvati Nehaj ϕ 0 0 displaystyle phi 0 infty to 0 infty nbsp neperervna strogo zrostayucha funkciya Todi isnuye obernena funkciya ϕ 1 displaystyle phi 1 nbsp i todi dlya vsih dodatnih a displaystyle a nbsp i b displaystyle b nbsp a b 0 a ϕ x d x 0 b ϕ 1 y d y displaystyle ab leq int 0 a phi x dx int 0 b phi 1 y dy nbsp Nerivnist perehodit u rivnist todi i lishe todi yaksho b ϕ a displaystyle b phi a nbsp Dlya rozuminnya dovedennya dostatno prosto namalyuvati z dovilnoyu ϕ displaystyle phi nbsp Vlasne dovedennya Redaguvati Dovedennya nerivnosti Geldera pokladayetsya na takij fakt dlya vsih p 1 displaystyle p in 1 infty nbsp i dlya bud yakih dodatnih stalih a displaystyle a nbsp i b displaystyle b nbsp a b a p p b p p displaystyle ab leq frac a p p frac b p p nbsp 1 de 1 p 1 p 1 displaystyle frac 1 p frac 1 p 1 nbsp tobto p p p 1 displaystyle p frac p p 1 nbsp Dlya p p 2 displaystyle p p 2 nbsp nerivnist ochevidna oskilki a b 2 0 displaystyle a b 2 geq 0 nbsp i zvidsi a 2 2 a b b 2 0 displaystyle a 2 2ab b 2 geq 0 nbsp z cogo a b a 2 2 b 2 2 displaystyle ab leq frac a 2 2 frac b 2 2 nbsp Dovedemo nerivnist u zagalnomu vipadku Vikoristayemo lemu navedenu vishe Vizmimo ϕ x x p 1 displaystyle phi x x p 1 nbsp Oskilki p gt 1 displaystyle p gt 1 nbsp mayemo ϕ 0 0 displaystyle phi 0 0 nbsp i ϕ displaystyle phi nbsp ye neperervnoyu i strogo vishidnoyu funkciyeyu Otzhe ϕ 1 y y 1 p 1 displaystyle phi 1 y y frac 1 p 1 nbsp i z lemi mi otrimuyemo a b 0 a x p 1 d x 0 b y 1 p 1 d y a p p b p p displaystyle ab leq int 0 a x p 1 dx int 0 b y frac 1 p 1 dy frac a p p frac b p p nbsp Vidno sho nerivnist perehodit u rivnist todi i lishe todi koli b a p 1 displaystyle b a p 1 nbsp sho totozhno do b p a p p 1 a p displaystyle b p a p p 1 a p nbsp Poklademo a x i d p x 0 displaystyle a frac x i d p x 0 nbsp i b y i d p y 0 displaystyle b frac y i d p y 0 nbsp Zavdyaki 1 mi znahodimo x i y i d p x 0 d p y 0 x i p p d p x 0 p y i p p d p y 0 p displaystyle frac x i y i d p x 0 d p y 0 leq frac x i p p d p x 0 p frac y i p p d p y 0 p nbsp i zvidsi beruchi sumu po vsih i displaystyle i nbsp vid 1 do n displaystyle n nbsp S i 1 n x i y i d p x 0 d p y 0 S i 1 n x i p p d p x 0 p S i 1 n y i p p d p y 0 p 1 p 1 p 1 displaystyle frac Sigma i 1 n x i y i d p x 0 d p y 0 leq frac Sigma i 1 n x i p p d p x 0 p frac Sigma i 1 n y i p p d p y 0 p frac 1 p frac 1 p 1 nbsp Otzhe S i 1 n x i y i d p x 0 d p y 0 displaystyle Sigma i 1 n x i y i leq d p x 0 d p y 0 nbsp sho i potribno bulo dovesti Chastkovi vipadki RedaguvatiNerivnist Koshi Bunyakovskogo Redaguvati Poklavshi p q 2 displaystyle p q 2 nbsp otrimuyemo Nerivnist Koshi Bunyakovskogo dlya prostoru L 2 displaystyle L 2 nbsp Evklidiv prostir Redaguvati Rozglyanemo Evklidiv prostir E R n displaystyle E mathbb R n nbsp abo C n displaystyle mathbb C n nbsp L p displaystyle L p nbsp norma u comu prostori maye viglyad x p i 1 n x i p 1 p x x 1 x n displaystyle x p left sum limits i 1 n x i p right 1 p x x 1 ldots x n top nbsp todi i 1 n x i y i i 1 n x i p 1 p i 1 n y i q 1 q x y E displaystyle sum limits i 1 n x i cdot y i leq left sum limits i 1 n x i p right 1 p cdot left sum limits i 1 n y i q right 1 q quad forall x y in E nbsp Prostir lp Redaguvati Nehaj X N F 2 N m displaystyle X mathbb N mathcal F 2 mathbb N m nbsp skinchenna mira na N displaystyle mathbb N nbsp Todi mnozhina vsih poslidovnostej x n n 1 displaystyle x n n 1 infty nbsp takih sho x p i 1 x n p lt displaystyle x p sum i 1 infty x n p lt infty nbsp nazivayetsya l p displaystyle l p nbsp Nerivnist Geldera dlya cogo prostoru maye viglyad n 1 x n y n n 1 x n p 1 p n 1 y n q 1 q x l p y l q displaystyle sum limits n 1 infty x n cdot y n leq left sum limits n 1 infty x n p right 1 p cdot left sum limits n 1 infty y n q right 1 q quad forall x in l p y in l q nbsp Jmovirnisnij prostir Redaguvati Nehaj W F P displaystyle Omega mathcal F mathbb P nbsp jmovirnisnij prostir Todi L p W F P displaystyle L p Omega mathcal F mathbb P nbsp skladayetsya z vipadkovih velichin iz skinchennim p displaystyle p nbsp m momentom E X p lt displaystyle mathbb E left X p right lt infty nbsp de simvol E displaystyle mathbb E nbsp poznachaye matematichne spodivannya Nerivnist Geldera v comu vipadku maye viglyad E X Y E X p 1 p E Y q 1 q X L p Y L q displaystyle mathbb E XY leq left mathbb E X p right 1 p cdot left mathbb E Y q right 1 q quad forall X in L p Y in L q nbsp Div takozh Redaguvatiprostir Lp Gelder Otto Nerivnist Yunga Nerivnist MinkovskogoDzherela RedaguvatiBekkenbah E Bellman R Neravenstva Moskva Nauka 1965 ros Otrimano z https uk wikipedia org w index php title Nerivnist Geldera amp oldid 40328711
