Basic Results in Functional Analysis
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1 Preared by: F.. ewis Udaed: Suday, Augus 7, 4 Basic Resuls i Fucioal Aalysis f ( ): X Y is coiuous o X if X, (, ) z f( z) f( ) f ( ): X Y is uiformly coiuous o X if i is coiuous ad ( ) does o deed o. f ( ): X Y is ischiz if f ( z) f( ) z ocally if for every, z i a comac se, vs. Globally if for all, z. Derivaive of f() a is f ( z) f( ) f '( ) lim z z f() is differeiable a if derivaive eiss a. he f ( ) C. f() is coiuously differeiable a if derivaive eiss a ad is coiuous. he f ( ) C. f() is uiformly coiuous if i is coiuous ad is derivaive is bouded. ischiz imlies uif. co. Coi. diff. imlies locally ischiz e f() be coi diff o X. he globally ischiz imlies f '( ), for a cos, X Diff imlies co e f( ) f() coi imlies eiss soluio () f() coi ad ischiz imlies eiss a uique soluio () ocally o a comac se, or globally. f() coi diff imlies eiss a uique soluio ()
2 Def. A ier roduc o a liear vecor sace X wih field F is a fucio.,. : X X F such ha for, yz, X, F a., b., iff c., y, y, homogeeous d., yz, y, z, liear e., y y,, comle cojugae Usually.,. : R R R. Def. A orm o R is a fucio.: R R such ha, for, y R, R: a. b. iff c., homogeeiy d. y y, riagle iequaliy Fac. Every orm is a cove fucio. Fac. Every ier roduc defies a orm /, Def. A semiorm or seudoorm does o have roery b. Def. A quasiorm has d. relaced by he milder roery y K y, K Aoher Def. A quasiorm has c. relaced by he milder roery Def. vecor -orm (Holder orms) for i i / R wih comoes i R Fac. k k, for some k i. his meas all vecor orms o R are equivale.
3 Mikowski iequaliy or riagle iequaliy y y Holder s iequaliy. e, y R ad q. he, y y q Cauchy-Schwarz iequaliy is he secial case =q=, y y Sylveser s iequaliy mi ( A) A ma ( A) wih he sigular values of A. Def. Covergece of sequeces. A sequece of vecors { k},, X is said o coverge o a limi vecor if k as k or equivalely, N k N k Def. is a accumulaio oi of sequece { k } if here is a subsequece of { k } ha coverges o. ha is, here is a ifiie subse K of oegaive iegers such ha { k} coverges o. k K Def. A se S X is closed if ad oly if every coverge sequece wih elemes i S has a limi i S. Def. A sequece { k } X is said o be a Cauchy sequece if as k, m k m Fac. Every coverge sequece is Cauchy, bu o vice versa. Def. Baach Sace. A ormed liear sace X is comlee if every Cauchy sequece coverges o a vecor i X. A comlee ormed liear sace is a Baach Sace. Def. A re-hilber Sace is a liear sace X wih a ier roduc. Def. Hilber Sace. A liear sace X wih ier roduc is comlee if every Cauchy sequece coverges o a vecor i X. A comlee liear sace wih ier roduc is a Hilber Sace. Fac. A Hilber sace has a ier roduc, hece a orm, hece is a Baach Sace. 3
4 Fac. X orm is R is a Hilber Sace wih ier roduc. y y,, y R. he associaed / ( ), he vecor orm. Fac. A bouded sequece { k } i R has a leas oe accumulaio oi i Fac. A sequece of real umbers { r k } which is moooically odecreasig ad bouded from above, i.e. rk rk R, R cos coverges o a real umber Fac. A sequece of real umbers { r k } which is moooically oicreasig ad bouded from below, i.e. rk rk R, R cos coverges o a real umber R. Coracio Maig. e S be a closed subse of a Baach sace X ad le be a maig ha mas S io S. Suose ha ( ) ( y) y,, ys, he * * * a. here eiss a uique vecor S such ha ( ). (fied oi) * b. ca be obaied by he mehod of successive aroimaio sarig from ay iiial oi i S. Fac. e f ( ) : R R. he f () d f() d his is acually he riagle iequaliy, for oe ha he ebesgue iegral is So ha b b / f () d lim f( k) k b b / b / f () d lim f( k) lim f( k) k k ier roduc. f, g f ( ) g( ) d D 4
5 ca deoe ime ad D a ime ierval, or ca ake ad iegrae over a regio D R. Def. orm (also deoed orm) of fucio f ( ) : R Y f(.) f( ) d, also deoed as f. D ca deoe ime ad D a ime ierval, or ca ake / ad iegrae over a regio D R. Def. Uiform fucio orm, or suremum orm, or orm, or Chebyshev orm f su f( ) : R Def. f is said o belog o if f (.) is bouded. Def. he ebesque ormed sace { f(.) Y : f(.) } (also deoed ) is If ime iegral, he ier roduc of f ( ), g( ) : R R is f, g f ( ) g( ) d ad orm of fucio f ( ) : R R is / f (.) f ( ) d, If over ime f su f( ) : Defie ier roduc f, g, f ( ) g( ) d ad orm f (.) f ( ) d, Def. he eeded ebesque ormed sace e (also deoed { f(.) Y : f(.), } e / e ) is 5
6 m Mea Value heorem. e f ( ) : R R be differeiable a each oi R, a oe se. e y, wih he lie segme y (, ). he here eiss a oi z of y (, ) such ha f f( y) f( ) y z m Imlici Fucio heorem. e f ( y, ) : RR R be coiuously differeiable a each m oi (, y) R R, a oe se. e (, y) such ha f(, y) ad f Jacobia mari is osigular. (, y) m he here eis eighborhoods U R of ad V R of y such ha y V he equaio f(, y) has a uique soluio U. Moreover he soluio ca be wrie as g( y) where g(.) is coiuously differeiable a y=y. eibiz Formula. () If () f(,) d he () () d ( ) '() f( ( ),,) '( ) f( ( ),,) '( ) f(, ) d d Bellma-Growall emma. e here be coiuous fucios, : R R, ad a coiuous oegaive fucio : R R. If he () () ()() s s ds, ( ) d s () () () s () s e ds, Also, if is a cos he () e, ( ) d If is a cos he ( () e ), () (Growall emma) 6
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