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First Applications PART C APPLICATIONS x FIRST APPLICATIONS In this paragraph we list a numb er of applications which can b e derived in a rather straightforward manner from the main selection theorems see Theory Among them are Banach valued version of the Dugundji extension theorem the Bartle Graves theorem Kadecs solution of the homeomorphism problem for Banach spaces the Mazurkiewicz theorem paracompactness of CWcomplexes on continuous choice in the continuity typ e denitions etc Extension theorems Recall the Dugundji extension theorem Theorem Let A be a closed subset of a metrizable space X and f A E a continuous mapping of A into a local ly convex topological linear space E Then there exists a continuous mapping f X E such that f j f Moreover one can assume that f X convf A A Using Convex valued selection theorem one can obtain a version of Theo rem with weaker restrictions on the domain and with stronger hyp othe ses on the range of the continuous singlevalued mapping f Theorem Let A be a closed subset of a paracompact space X and f A B a continuous mapping of A into a Banach space B Then there exists a continuous mapping f X B such that f j f Moreover one A convf A can assume that f X convf A Then C is a closed convex subset of the Proof Let C Banach space B Dene a multivalued map F X B by C x A F x ff xg x A Clearly Convex valued selection theorem can be applied to F to get the desired continuous extension f X B of f Note that for the intersection of assertions of Theorems and ie for metrizable domains and Banach spaces as ranges we practically have two pro ofs of the same extension result The only dierence is that f X convf A in Theorem whereas f X convf A in Theorem In the rst case the values of f x can be found directly via some BartleGraves type theorems Theory of liftings suitable lo cally nite partition of unity Theorem On the other hand the second pro of gives no straightforward answer for f x because it uses an inductive pro cess of a construction of selections of the multivalued n mapping F Theorem We also note that for separable Banach spaces for example the real line IR theorems of Dugundji Hanner and Urysohn yield stronger results Theorem A T space X is normal if and only if every continu ous mapping f A B of a closed subset A X into a separable Banach space B admits an extension f X B Note that Convex valued selection theorem and its converse gives a characterization of paracompactness but the converse of Theorem gives an unknown class of a top ological spaces It is only known that in the assertions of Theorem it is p ossible to pass outside the class of metrizable spaces and consider the so called stratiable spaces see BartleGraves typ e theorems Theory of liftings We b egin by a standard fact from linear algebra If L is a linear op erator from a nitedimensional vector space X onto a vector space Y then X is isomorphic to the direct sum Y Ker L of the range Y and the kernel Ker L of the op erator L This is also true for a nitedimensional space Y and an arbitrary lo cally convex top ological vector space X The op erator and the isomorphism X Y Ker L in the last case must of course b e continuous or b ounded see Lemma However the situation is quite dierent when X and Y are b oth innite dimensional top ological vector spaces In fact every separable Banachspace Y is the image of the Banach space l of all summable sequences of real numb ers under some continuous linear surjection L If we supp ose that l is isomorphic to the direct sum Y Ker l then the space Y b ecomes isomorphic to a complementable subspace of the space l However by all innite dimensional complementable subspaces of the space l are isomorphic to l Hence we would prove that all innitedimensional separable Banach spaces are isomorphic to l Contradiction Therefore thereisasa rule no isomorphism b etween X and Y Ker L for innitedimensional spaces X and Y and for continuous linear surjections L X Y However a homeo morphism between X and Y Ker L always exists This is the content of the Bartle Graves theorem Theorem Let L be a continuous linear operator which maps a Banach space X onto a Banach space Y Then there exists a section f of L ie a continuous mapping f Y X such that L f idj Moreover Y there exists a homeomorphism between X and the direct sum Y Ker L First applications Proof By the Banach op en mapping principle L is an op en mapping ie it maps op en sets to op en sets Hence F L Y X is a lower semicontinuous multivalued mapping and the values F y y Y are convex closed subsets of X b ecause they are parallel translates of the kernel Ker L L of the op erator L So let f Y X b e a selection of F guaranteed by Theorem A Dene g Y X by the formula g y f y f y Y Then g y is also an elementof F y L y since f L f y L y andLg y Lf y Lf Lf y y So g Y X is also a selection of F and g One can now dene h X Y Ker L by the equality hxLxx g Lx The continuity of h follows by the continuity of L and g It follows from Lx g Lx Lx L g Lx that hx indeed lies in Y Ker L If x y and x y Ker L then hx hy b ecause hx hy Lx y x y g Lx g Ly If x y and x y Ker Lthen x y and LxLy So on the right hand side of the last equality the rst co ordiante is zero and the second one is nonzero Therefore wecheck injectivity of h Toprove that h is onto it suces to note that for every y z Y Ker L hg y z Lg y gy z g Lg y Lz y z To complete the pro of note that the inverse mapping h is given by the formula h y z g y z and it is continuous b ecause of the continuity of g Theorem holds for everypairX Y of spaces to which the Banach op en mapping principle and Selection theorem A apply For example it is also true for Frechet spaces ie for completely metrizable and lo cally con vex top ological vector spaces Moreover one can consider completely metriz able nonlo cally convex X and Y but add the lo cally convexity assumption for the kernel Ker L see or Prop osition I I Note that the pro of of Convex valued selection theorem via Zerodimensional selection theorem see Theory x gives the third pro of of such a generalization of Theorem Indeed it suces to observe that every continuous function over com pactum into Ker L is integrable b ecause of the lo cal convexity of Ker L Theorem admits the following interesting renement which says that a section for L can b e almost linear Theorem Under the hypotheses of Theorem there exists a section s Y X of L such that sty tsy for al l t IR and y Y BartleGraves type theorems Theory of liftings Proof Let g Y X b e a section of L such that g see the pro of of Theorem Let ky k g yky k y hy y Then h is a section of L due to the linearity of L and hty thy for every t To complete the pro of it suces to set sy hy hy y Y For spaces over the eld of complex numbers C Theorem is still valid but with a more complicated formula for s Z sy zh zy dz y Y S where S fz C jkz k g and dz is the usual invariant normalized mea sure on S Only the additive prop erty sx y sxsy fails to hold for s However this is a key p oint b ecause a linear section s would yield a contra diction as it was explained at the b eginning of this section We end this section by a formulation of the parametric version of The orem The pro of is more sophisticated than the previous ones see Theorem Let X and Y b e Banach spaces and supp ose that the space EndX Y of all linear continuous op erators whichmapX onto Y is nonempty Weequip EndX Y with the usual supnorm kLk supfkLxk jkxk g For Y X every L EndX Y and for every y Y letdL y b e the distance b etween the origin X and the hyp erspace L y ie dL y inf fkxkjx L y g Theorem For every there exists a continuous map ping f EndX Y Y X such that for every pair L y EndX Y Y a f L y L y ie f L is a section of L b kf L y k dL y and c f L y f L y for al l IR IRnfg In summary Theorem states that in Theorems and a section f of a linear continuous surjection L can be chosen to continuously dep end on this surjection The additional statement b shows that the values of the selection f L canbechosen to lie at the minimal p ossible distance from the origin X First applications Note that without condition c it is p ossible to reduce Theorem to the standard Convex valued selection theorem A In fact for a xed one can consider a multivalued mapping F EndX Y Y X dened by F L y L y Cl D dL y Then the values of F are nonempty closed convex subsets of X After a verication of lower semicontinuityof F we can obtain f as a selection of F For a sp ecial endomorphism L between sp ecial Banach spaces the prob lem of nding linear selections for L admits a solution without using the selection theory techniques So let X A be a measure space and L L X A a Banach space in fact algebra of all b ounded measurable functions f X IR with p ointwise dened vector op erations Let P L L b e the quotient mapping which asso ciates to every f L its equivalence class ie the set of all functions equivalent to f Then L L X A is the Banach space in fact an algebra endowed with the following norm kf k ess supfkf xkjx X g So a homomorphism L L of Banach algebras L and L is said to be a lifting if is a selection of P and id j id j In summary X X existence of lifting gives a way to talk ab out values of elements of the space L at p oints x X If is a linear not algebraically
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