On the Hahn-Banach Theorem and Its Application to Polynomial and Dirichlet Problems

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1 O the Hah-Baach Theorem ad Its Alcato to Polyomal ad Drchlet Problems Emmauel Okereke, Ph.D. Abstract Eteso osed a serous mathematcal roblem fuctoal aalyss utl Hah ad Baach came u wth the effcet Hah-Baach theorem for eteso of fuctoals Baach sace. Subsequetly other eteso results have cotued to evolve eve toology ad fuctoal aalyss. I secto two of ths work, the bascs of these results were cosdered recsely as theorems ad roostos. These strogly gave rse to the backgroud of the alcatos cosdered ths work as secto three.. INTRODUCTION Kowg that a lear fucto o a vector sace X s a lear oerator from X to the sace R of real umbers ad that a lear fuctoal s a real valued fucto o X such that fα by αf bfy, we the ask a frst questo o how we ca eted a lear oerator or fuctoal from such a sace to the whole sace X such a way that the varous roertes of the fuctoal are reserved. I vew of ths, a fudametal eteso theorem, the Hah- Baach theorem ad ts cosequet corollares relmarly came to lay etesvely secto two, the et secto. As secod questo, we assume that the fucto g s gve for oly a subset X of the real sace, C, ca we eted X so as to defe a lear fuctoal g of orm M g M the etre sace C. I vew of ths, a evdet ecessary codto s that for all lear combatos of elemets of X, we have ck gf k M ckf k.... Joural of Mathematcal Sceces & Mathematcs Educato Vol. 9 No. 8

2 If we let M g to be the smallest value of M for whch. s fulflled, we observe that codto. becomes a suffcet codto for aswer to the above mmedate questo. 2. PRELIMINARY RESULTS ON EXTENSION Theorem 2.: Hah-Baach: let be a real- valued fucto defed a vector sace X satsfyg y y ad α α for each α 0. Suose that f s a lear fuctoal defed o a subsace S ad that fs s for all s S. the there s a lear fuctoal f defed o X such that f for all, ad fs fs for all s S. Proof: Cosder all ler fuctoal g defed o a subsace of X ad satsfyg g Wheever g s defed. Ths set s artally ordered by settg g < g 2 f g 2 s a eteso of g, that s, f the doma of g s cotaed the doma of g 2 ad g g 2 of the doma of g. By the Hausdorff Mamal Prcle there s a mamal learly ordered subfamly {g α } that cotas the gve fuctoal f. We defe a fuctoal whch s deedet of the α chose. The doma of f s a subsace ad f s a lear fuctoal, for f ad y are the doma of f. the doma g α ad y doma g β for some α, β. By the lear orderg of {g α }, we have ether g α < g β or g β < g α, say the former. The ad y are the doma of g β, ad so λ µy s the doma of g β ad so the doma of f, ad f λµy g β λµy λg β µg β y λf µfy. Thus f s a eteso of f. Moreover, f s a mamal eteso for f G s ay eteso of f, g α < f< G mles that G must belog to g α by the mamalty of g α. Hece G < f, ad so G f. It remas oly to show that f s defed for all X. sce f s mamal. Ths wll follow f we ca show that each g that s defed o a roer subsace T of X ad satsfes gt t has a roer eteso h. Joural of Mathematcal Sceces & Mathematcs Educato Vol. 9 No. 9

3 Let y be a elemet X T. we shall show that g may be eteded to the subsace U saed by T ad y, that s, to the subsace cosstg of elemets of the form λy t wth t T. f h s a eteso of g, we must have hλy t λhy ht λhy gt, ad so h s defed as soo as we secfy hy. for t, t 2, T we have gt gt 2 gt t 2 t t 2 t -y t 2 -y. Hece -t -y gt t 2 y-gt 2, ad so su t ε T [ t y g t ] f [ t y g t ]. t ε T Defe hy α, where α s a real umber such that su{-t-y gt} α f [ty-gt. we must show that hλyt λαgt λy t. f λ > 0, the λα gt λ{α gt/λ}. λ λ [{t/λ y-gt/λ} gt/λ] λt/λ y t λy. f λ - µ < 0, the α µ gt µ - α gt/µ µ {t/µ - y gt/µ} gt/µ µ t/µ -y t µy. Thus hλy t λy t for all λ, ad h s a roer eteso of g. The followg roosto s a geeralzato of the Hah- Baach Theorem whch s useful certa alcatos. By a Abela semgrou of lear oerators o a vector sace X, we mea a collecto G of lear oerators from X to X such that of A ad B are G, the AB BA ad AB s G. we also assume that the detty oerator belogs to G. Proosto 2.: let, s,, ad f be as Theorem 2., ad let G be all Abela semgrou of lear oerators o X such that for every A G we have A for all X, whle for each s S we have As S ad fas fs. the there s a eteso f of F to a lear fuctoal o X such that f ad FA for all X [0] Proof: defe a fucto q o X by settg Joural of Mathematcal Sceces & Mathematcs Educato Vol. 9 No. 0

4 Joural of Mathematcal Sceces & Mathematcs Educato Vol. 9 No. q f m A..A, Where the f s take over all fte sequeces {A,...A } from G. we clearly have q ad qα αq for α 0. for ay ad y X ad ay ars {A,...A } ad {B, B m } of fte sequeces from G we have. m m m m y B m A y B A m A B m y A B m y q Takg fma over every ar {A }, {B }, we obta q y q qy. Sce qθ θ 0, we have 0 q- q q- q - Thus q caot be -, ad q s real valued. For s S, s A A s A s A s f s f Hece fs qs, ad we may aly Theorem 2. wth relaced by q to obta a eteso f of F to all z such that f q. t remas oly to show that fa f. Now q-a... A A A A A ]. [ A Sce ths s true for each, we have q-a 0. sce f fa f-a q-a 0, we have fa, ad alyg ths to, we get f fa.

5 Proosto 2.2: let be a elemet a ormed vector sace X. The there s a bouded lear fuctoal f o X such that f f. [3] Proof: let S be the subsace cosstg of all multles of, ad defe f o S by fλ λ ad set y y. The by the Hah Baach Theorem there s a eteso of f to be a lear fuctoal o X such that fy y. Sce f-y y, we have f. Also f f.. Thus f ad f f.. Proosto 2.3: let T be a lear subsace of a ormed lear sace X ad Y Elemet of X whose dstace to T s at least δ, that s, ad elemet such that y t δ for all t ε T. the there s a bouded lear fuctoal f o wth f, fy δ, ad such that ft 0 for all t T. [5] Proof: let S be the subsace saed by T ad y, that s, the subsace cosstg of all elemets of the form αy t wth t ε T. Defe fαy t αδ, we have fs s o S. by the Hah Baach Theorem we may eted f to all of X so that f. But ths mles that f. By the defto of f o S, we have ft 0 for t T ad fy δ. The sace of bouded lear fuctoal o a ormed sace X s called the dual or cougate of X ad s deoted by X *. Sce R s comlete, the dual X * of ay ormed sace X s a Baach sace by Proosto 2.3. Two ormed vector saces are sad to be sometrcally somorhc f there s a oe to oe lear mag of oe of them to the other whch reserves orms. From a abstract ot of vew, sometcally somorhc saces are detcal, the somorhsm merely amoutg to a refet of the elemets. I suose we kow that the dual of L P was sometrcally somorhc to L q for < ad that there was a atural reresetato of the bouded lear fuctoal o L by elemets of L q. We are ow a osto to show that a smlar reresetato does ot hold for bouded lear fuctoal o L [0,]. We ote that C[0,] s a closed Joural of Mathematcal Sceces & Mathematcs Educato Vol. 9 No. 2

6 subsace of L [0,]. Let f be that lear fuctoal o C[0,] whch assgs to each C[0,} the value 0 of X at 0. It has orm o C, ad so ca be eteded to a bouded lear fuctoal f o L [0,]. Lear fuctoal ad Hah-Baach Theorem Gve L [0,] such that f 0 y dt for all C, let { } be a sequece of cotuous fuctos o [0,] that are bouded by, we have 0, ad such that t 0 for all t 0. The, for each y L, dy 0, whle f. If we cosder the dual ** of *, the to each X, there corresods a elemet ϕ ** defed by ϕf f. We have su f. Sce f f, we have ϕ, f. Whle by roosto 2.2 we have a f of orm wth f. Hece ϕ. Sce ϕ s clearly a lear mag, ϕ s a sometrc somorhsm of oto some lear subsace ϕ [] of **. The mag ϕ s called the atural somorhsm of oto **, ad f ϕ[] ** we say that s refleve. Thus L s refleve f < <. Sce there are fuctoals o L that are ot gve by tegrato wth resect to a fucto o l L, t follows that L s ot refleve. It should be observed that may be sometrc wth ** wthout beg refleve. By Proosto 2.3, the sace ** s comlete, ad so the closure ϕ[] vector sace s sometrcally somorhc to a dese subset of a Baach sace. Before closg ths secto, we add a word about the Hah Baach Theorem for comle vector saces. A comle vector sace s a vector sace whch we allow multlcato by comle scalars. The followg eteso of the Hah-Baach Theorem for comle saces s due to Boheblust ad Sobczyk: Joural of Mathematcal Sceces & Mathematcs Educato Vol. 9 No. 3

7 Theorem 2.2: let X be a comle vector sace, S a lear subsace, a real valued fucto o X such that y y, ad α α. let f be a comle lear fuctoal o S such that fs s for all s S. The there s a lear fuctoal f defed o such that fs fs for s S ad f for all X. [6] Proof: we frst ote that ca be cosdered as a real vector sace f we smly gore the ossblty of multlyg by comle costats. A mag f from X to the comle umbers that s lear the real sese s lear the comle sese f a oly f f f for each. o S defe g ad h by takg gs to be the real art of fs ad hs the magary art. The g ad h are lear the real sese ad f f h. Sce f s lear the comle sese, gs hs fs gs hs, ad so hs - gs. Sce gs fs s, we ca eted g to a fuctoal g o X that s lear the real sese ad satsfes G. let F G G. The Fs for s S. sce F G G [G G], we have F lear the comle sese. For ay X, choose ω to be a comle umber such that ω ad ωf F. The F ωf Fω Gω ω. 3. APPLICATIONS 3.. THE DIRICHLET PROBLEM If Ω be a bouded doma R ad let 2 2 /..the equato u s called the Lalace equato. Ay fucto C 2 Ω that satsfes the Lalace equato Ω s called a harmoc fucto Ω. We shall deote by Ω the boudary of Ω. We say that Ω s C 2 f Ω ca be covered by a fte umber of oe subsets G, wth each G Ω havg a arametrc I Joural of Mathematcal Sceces & Mathematcs Educato Vol. 9 No. 4

8 reresetato terms of fuctos 2 C. The Drchlet s the followg oe: gve a cotuous fucto f o Ω cotuous Ω, ad satsfyg u 0 Ω, u f Theorem 3.. The Drchlet roblem has at most oe soluto. Ths follows mmedately from the followg result, kow as the weak mamum rcle. Theorem 3..2 Let u be a harmoc fucto Ω, cotuous Ω. The ma u mau [2] Ω Ω Proof. If the asserto s ot true, the there s a ot y Ω such that 2 u y > mau. cosder the fucto v u ε y f. ε > 0 s Ω > v suffcetly small, the v y u y ma s attaed at a ot z Ω. At the ot, Hece v v 0 u ε 2 v, at z.however, 2 z y 2ε > 0 Ω,... v a cotradcto as for the estece of a soluto to the Drchlet roblem, we quote the followg result: Theorem 3..3 If Ω s C 2, the for ay cotuous fucto f o Ω there ests a soluto to the Drchlet roblem 3.., [2] Theorem Is much deeer tha Theorem 3... There are varous methods of rovdg t, but each of them requres legthy develomets. Cosder ow the fucto Joural of Mathematcal Sceces & Mathematcs Educato Vol. 9 No. 5

9 2 y, k, y log y, f > 3, f 2. We call t a fudametal soluto of the Lalace equato. It satsfes the Lalace equato, whe vares Ω {y}, ad t grows to as y. [7] Defto 3.. A fucto G, y, defed for each y Ω ad Ω y, s called Gree s fucto for the Lalace oerator Ω f: G, y k, y h, y, where h, y whe Ω G, y s harmoc s cotuous whe Ω [y]. G, y 0 for Ω. [2] Oe ca use Gree s fucto order to rereset the soluto u of the Drchlet roblem 3.., 3..2 terms of a tegral volvg the boudary values f. What we shall ow do s costruct Gree s fucto by usg, as a tool, the Hah-Baach theorem. We shall cosder oly the case 2, whch s somewhat smler tha the case where 3. we shall assume that Ω cossts of a defte umber of cotuously dfferetable closed curves. The followg codto the holds the roof s omtted: P For ay z ear Ω, deote by T the taget lae to Ω Ω earest to z. deote by z the reflecto of z wth resect to T the at the ot o Joural of Mathematcal Sceces & Mathematcs Educato Vol. 9 No. 6

10 ma Ω z z' f z z 0, z 0 Ω Theorem 3..4 f 2 ad Ω s C, the Gree s fucto ests. Proof. Deote by the Baach sace of all cotuous fuctos o Ω wth the mamum orm, ad deote by the lear subsace cosstg of those fuctos f for whch the Drchlet roblem 3.., 3..2 has a soluto. For a y Ω, cosder the lear fuctoal L y o X defed by L y f uy, where u s the soluto of 3.., From Theorem 3..2 t follows that L y s bouded ad that ts orm s. By the Hah- Baach theorem L y ca be eteded to a bouded lear fuctoal o X, havg orm. We deote such a eteso aga by L y. For each z Ω cosder the elemet f z log z Ω ad defe k z L f. We y y z clam that k y z s a harmoc fucto. To rove t let z z k y z' k δ, z. ' 2 δ y the z L y f z f δ z As δ 0, f z f z / δ f z / z, where f z / z s the elemets log / z of. sce L y s a cotuous oerator o X, we get k y z z ests ad equals 3.2. SPACES OF POLYNOMIALS 3.2. Itroducto fz z We kow that ordarly t s mossble to eted scalar valued cotuous homogeous olyomals over a larger Baach sace usg the oular Hah- Joural of Mathematcal Sceces & Mathematcs Educato Vol. 9 No. 7 L y

11 Baach theorem uless by the use of a more rgorous but a stroger aroach whch volves to a etet the use of a ultra ower method whch ca be related to the estece of such eteso morhsm by comlemetg X** G** f: G X* whe a X**. Ths s ust eactly the same as the estece of a lear eteso morhsm for cotuous homogeous olyomal k F k G ad our terest s the erfect Hah-Baach tye eteso wth suffcet codto for the estece of each ad ot a lear morhsm that wll eted all of them. We do ths by detfyg olyomals more so, we kow that the sace k X of K- homogeous olyomals over a Baach sace X s the dual sace of the sace of symmetrc ktesor of elemets of X wth the roectve toology k X beg the dual of a sace saed by evaluatos. Normed wth lear fuctoal sut that we use Hah- Baach theorem to eted them. We cosder the followg before we delver to the ma alcato. Gve a elemet X, at s a cotuous lear form over the sace of K-homogeous olyomals we deote the evaluato morhsm by e. The orm of e as a elemet of k X s k. deoted by S the lear subsace of k X saed by all evaluatos at ots of X. Ths s a o-closed subsace whose elemets have o- uque reresetatos of the form s e, For comle X the orm of such a elemet s s su P : P P k X of orm oe whch s deedet of the reresetato of s. Whe s a real Baach sace ad k s eve, the reresetato must take to accout the sgs, so s m e e y. as our results are vald for both the real ad comle case, but we wll use oly the comle-case otato. Note also that for ay scalar λ, e λ λ k e. The dual of s ca be readly see to be k X. Joural of Mathematcal Sceces & Mathematcs Educato Vol. 9 No. 8

12 We wll eed to cosder other toologes o the vector sace s. we kow from 4 that the Borel trasform B: k X k X s the lear ma gve by BT γ k. A lear form T s the kerel of the B f ad oly f T s zero over the sace of aromable olyomals. Sce geerally ot all olyomals are aromable, B s rarely oe to oe. However, the followg lemma assures that whe restrcted to the sace S, the Borel trasform s always oe-to-oe. Lemma[3.2.]: let,...,, f k k 0 for all ', the 0 forall P P. γ γ k X. Proosto 3.2.: S s sometrcally somorhc to P k X. Note that a mmedate cosequece of the roosto s that the mage of the Borel Ma : P k P k s the sace of tegral k-homogeeous olyomals over. Although we are maly cocered wth tegral olyomals, we wll take the mag T P T where P T Te as theorem, s a somorhsm betwee the algebrac dual of S ad the sace of all ot ecessarly cotuous k- homogeeous olyomals over X we wll deote ts reverse P T. Imosg more or less strget cotuty codtos o T wll roduce dfferet kds of olyomals. Hece we show that all saces of olyomals are roduced ths way. Proosto 3.2: If Z s ay subsace of P k cotag the fte tye olyomals, Z s algebracally somorhc to s,τ, where T s a Hausdorff cove toology o S.. Eteso Of Polyomals I ths secto, we cosder the roblem of etedg a olyomal defed o a Baach sace X to a larger Baach sace Y. thus, we wll use the learzato of dfferet tyes of olyomals reseted secto 3. coucto wth the Hah-Baach eteso theorem for locally cove, to roduce etesos of Joural of Mathematcal Sceces & Mathematcs Educato Vol. 9 No. 9

13 olyomals. We wll detfy ths secto the lear fuctoal TP wth the olyomal P. thus we may wrte P s,τ. Deote wth S X ad S Y the saces saed, as the roceedg secto, by evaluatos ots of X ad of Y. resectvely. The the cluso ma τ from X to Y duces a ma from S X to S Y. e e τ, Whch s oe-to-oe, thaks to the Lemma secto 3., ad the Hah-Baach theorem. Ideed, f e τ 0 S X, ad γ X, the eted γ to Γ ε Y, ad we have e γ e τ Γ 0, for ay γ X. Thus e 0. we wll dro the τ the sequel ad cosder ay s S X a elemet of S Y. Note that ay P S X * the algebrac dual of S X eteds to S Y, so ay olyomal ca be algebracally eteded; the real questo s what tye of olyomal we ca eect ths eteso to be. Now cosder locally cove, Hausdorff toologes τ ad τy o S X ad S y. The the followg roosto s mmedate. Proosto 3.2.: gve the saces S, τ y ad S y τ y of olyomal over X ad Y resectvely, the a olyomal P S X, τ eteds to a olyomals S y, τ y f t s τ G -cotuous.e. for the toology duced by τ y o S. [2] We ow eted tegral; olyomals over E to tegral olyomals over a arbtrary larger sace X. Theorem: Ay P P I k E ca be eteded to P P I k G, wth P I P I [4]. Corollary 3.2.: There s a etedble, o- tegral olyomal over c o. [2] Joural of Mathematcal Sceces & Mathematcs Educato Vol. 9 No. 20

14 To see ths, cosder a o- weakly comact symmetrc oerator T: l l, ad cosder the cluso l C[0,]. T corresods to a 2- homogeeous olyomal P overl, ad ts eteso to C[0,] would gve rse to a cotuous symmetrc oerator S: C [0,] C [0,] makg the followg dagram commutatve. l T l J J ' C [0,] s C [0,]' But ths ca ot be, for the symmetrc regularty of C [0,] assures the weak comactess of S, ad thus of T. [6] May sub-classes fte-tye, uclear of tegral olyomals over E ca of course be eteded to the corresodg kd of olyomal over G. we et state ths kd of corresodece for olyomals S E,τ 0. Proosto 3.2.: each P S E,τ 0 eteds to P S G,τ 0 ; [3]. Alyg these to the tegral olyomal, we kow that accordg to Aro ad Berer, ay cotuous homogeous olyomal ca be eteded from X to X whch dcates that eteso s fact urely algebraes ature ad caot be aled to ay olyomal cotrbutos hece we verfy that P s uclear olyomal over X. the ts Aro-Berer eteso P s also uclear. I fact, f P Σ I γ, the P Σ I wth γ γ where X X s the caocal cluso f P s tegral ad µ s a regular Borel o B E reresetg P. the oe s temted to ut for λ X. P z k z γ dµ B X where as the oly roblem wth ths eresso s that ths tegral may ot est. I fact z k eed ot be a µ- measurable fucto. Note that z k s ot a γ Joural of Mathematcal Sceces & Mathematcs Educato Vol. 9 No. 2

15 cotuous fucto o B E w*, or s the ot wse lmt of a sequece of owers of elemets of E whch are kow to be tegrable. [] We wll rove that the valdty of the above eresso for the Aro- Berer eteso of a tegral olyomal s equvalet to E ot cotag a somorhc coy of l. for ths we wll use the equvalece gve by Haydo [9], ad the characterzato of the Aro- Berer eteso []. We wll also be able to rove that the Aro-Berer eteso of a tegral olyomal s always tegral, wth the same tegral orm eve whe the above eresso s ot vald. µ wll deote a regular Borel measures o B,ω, ad k a ostve teger. Defe S, S : X l µ ad cosder l µ X as Sf f γ γ dµ γ B E l µ. It easly follows that s s ad, for ε X, s, where s the class of the fucto γ γ. Lemma 3.2.: If P s a tegral olyomal over X, rereseted by the measureµ, the ts Aro-Berer eteso P may be wrtte P z S' z k dµ [4] B E ' Corollary 3.2.2: The Aro- Berer eteso of a tegral olyomal s a tegral olyomal ad has the same tegral orm. [7] Proof: A olyomal Q o a Baach sace s tegral f ad oly f there ests a fte measure v o a comact sace Ω ad a bouded lear oerator R: l Ω such that Q Ω R k dv see[0]. Moreover, we have Q I R k v. our case, the revous lemma gves such a factorzato ad, sce s, P I P I. the other equalty s a cosequece of our frst roosto of l, sce s a eteso of as a lear fuctoal. Emmauel Okereke, Ph.D., Mchael Okara Uversty of Agrculture, Ngera Joural of Mathematcal Sceces & Mathematcs Educato Vol. 9 No. 22

16 Refereces [] Okemags, Bull. Soc. Math. Frace , [2] Aver, Fredma, foudatos of Moder Aalyss dover ublcatos c. New York, 982. [3] Baach, S. Theores deal oeratos Leares, 2 d Edto. Rert New York. Chelsea Publshg Co.975 [4] Bartle, R. ad Graves, L. Mags betwee fucto saces, Tras. Amer. Math. Soc , [5] Dave, A. ad Gamel, T. A theorem o olyomal- star aromato, Proc. Amer. Math. Soc , [6] Day, M. M. Normed Lear saces, 3 rd Edto Ergebsseder Mathematc ad Ihrercrezgebete N0 2 srger verlag, 973. [7] Dee, S. ad Tmoey, R. Comle geodesscs o cove domas, Progress Fuctoal Aalyss ed. Bersted, Boet, Horvath, Maestre, Math. Studes 70, North-Hollad, Amsterdam 992, [8] Galdo, P., Garca, D. M. Maestre, ad Muca, J. Eteso of multlear mags o Baach saces, to aear. [9] Guta, C. Malgrage theorem for uclearly etre fuctos of bouded tye o a Baach sace, Ph.D. Thess I.M.P.A., Ro de Jaero ad Uversty of Rochester 966. [0] Haydo, R. Some more characterzatos of Baach saces cotag l, Math. Proc. Camb. Phl. Soc , [] Hlle, E. ad Phls, R. S. fuctoal Aalyss ad Seor grous, revsed ad Clolloqucum ublcato Vol. 3 Provdece Amerca Mathematcal Socety [2] Jaramllo, J. A., Preto, A. ad Zalduedo, I. The bdual of the sace of olyomals o a Baach sace, Math, Proc. Camb. Phl. Soc. too aear [3] Krwa, P. ad Rya, R. Etedblty of homogeeous olyomals o Baach saces, to aear. Joural of Mathematcal Sceces & Mathematcs Educato Vol. 9 No. 23

17 [4] Ldestraauses, J. Eteso of comact oerators, Mem. Amer. Math.Soc Moraes, L. A Hah-BAach eteso theorem for some holomorhc fuctos, Comle Aalyss Fuctoal Aalyss ad Aromato Theory ed. J. Muca, Math Studes 25, North-Hollad. Amsterdam 986, [5] Muca, J. Learzato of bouded holomorhc mags o Baach saces, tras. Amer. Math. Soc , [6] Oford N. ad Schwartz J. T., Lear Oerators, Part, New York, Wley- Iterscece 958. [7] Rez F. ad Nagy B.. Fuctoal Aalyss, Eglsh Edto, New York Ugar, 956. Joural of Mathematcal Sceces & Mathematcs Educato Vol. 9 No. 24