Computability of the Hahn-Banach Theorem Revisited

Computabilit y of the Hahn-Banac h Theorem Revisited V asco Brattk a 1 , 2 [0000-0003-4664-2183] and Christopher Sorg 1 [0009-0003-9684-6966] 1 F akultät für Informatik, Universität der Bundesw ehr Münc hen, W erner-Heisenberg-W eg 39, 85577 Neubiberg, German y 2 Departmen t of Mathematics and Applied Mathematics, Universit y of Cape T o wn, Priv ate Bag X3, Rondeb osc h 7701, South Africa Vasco.Brattka@cca-net.de chr.sorg@unibw.de Abstract. Computational prop erties of the Hahn-Banac h theorem ha ve b een studied in computable, constructiv e and reverse mathematics and in all these approac hes the theorem is equiv alent to weak Kőnig’s lemma. Gherardi and Marcone pro ved that this is also true in the uniform sense of W eihrauc h complexity . How ever, their result requires the underlying space to b e v ariable. W e prov e that the Hahn-Banac h theorem attains its full complexity already for the Banach space ℓ 1 . W e also pro ve that the one-step Hahn-Banach theorem for this space is W eihrauch equiv alent to the in termediate v alue theorem. This also yields a new and very simple pro of of the reduction of the Hahn-Banac h theorem to w eak K őnig’s lemma using infinite pro ducts. Finally , we show that the Hahn-Banach theorem for ℓ 1 in the t w o-dimensional case is W eihrauch equiv alen t to the lesser limited principle of omniscience. Keyw ords: Computable analysis · W eihrauc h complexit y · Hahn-Banac h Theorem. 1 In tro duction The Hahn-Banac h theorem ( HBT ) is one of the core theorems of functional anal- ysis. Here and in the following w e consider only normed spaces ov er the field R and all computational results are for separable spaces. By ∥ f ∥ := sup ∥ x ∥≤ 1 | f ( x ) | w e denote the supr emum norm of a functional f in a normed space X . Theorem 1 (Hahn-Banach). L et X b e a norme d sp ac e with a line ar subsp ac e A ⊆ X . Then every line ar b ounde d functional f : A → R has a line ar extension g : X → R with ∥ g ∥ = ∥ f ∥ . A go od survey on the history of the Hahn-Banach theorem is pro vided b y Narici and Beck enstein [18]. The Hahn-Banac h theorem has been studied in computable analysis b y Metakides, Nero de and Shore [16,17], in reverse math- ematics by Brown and Simpson [12,19], and in reverse constructive analysis by 2 V. Brattk a and C. Sorg Ishihara [15,13]. In all these approaches the theorem turns out to b e equiv alen t to weak Kőnig’s lemma ( WKL ). Finally , Gherardi and Marcone [14] prov ed that this is also true in the uniform sense of W eihrauch complexity [8] (b elo w we will define W eihrauch equiv alence ≡ W and other relev ant notions used in the in tro duction). Theorem 2 (Gherardi and Marcone 2009). HBT ≡ W WKL . Using a Kleene tree, one obtains a result of Metakides, Nero de and Shore [17] as a corollary of this classification. Corollary 3 (Metakides, Nero de and Shore 1985). Ther e exists a c om- putable Banach sp ac e X with a c omputably sep ar able subsp ac e A ⊆ X and a c omputable functional f : A → R with ∥ f ∥ = 1 , such that f has no c omputable line ar extension g : X → R with ∥ g ∥ = 1 . Both aforementioned results work with the construction of a v ariable Banach space X that dep ends on the resp ectiv e instance of weak K őnig’s lemma. W e in vestigate the question whether there is a fixed computable Banach space X for whic h the Hahn-Banach theorem HBT X is W eihrauc h equiv alent to weak K őnig’s lemma. Ev ery singlev alued problem that maps to a computable metric spaces and is W eihrauch reducible to WKL is already computable [7, Theorem 8.8]. Hence, Theorem 2 implies that HBT X is computable if the computable normed space X has the prop ert y that all linear b ounded functionals on subspaces of X ha ve unique norm preserving extensions (this w as noted also in [3]). Due to a classical result of T aylor and F oguel [18, Theorem 16.4.8], it is kno wn that extensions are uniquely determined for a space X (for all subspaces and functionals) if and only if the dual s pace of X is strictly conv ex. Hence we obtain the follo wing. Corollary 4. HBT X is c omputable for every c omputable norme d sp ac e X with a strictly c onvex dual sp ac e. W e recall that the normed spaces ℓ p ( I ) := { x ∈ R I : ∥ x ∥ p < ∞} o ver coun table sets I are defined for 1 ≤ p ≤ ∞ by ∥ ( x i ) i ∈ I ∥ p := X i ∈ I | x i | p ! 1 p if p < ∞ and ∥ ( x i ) i ∈ I ∥ ∞ := sup i ∈ I | x i | . F or I = N we briefly write ℓ p and for I = { 0 , 1 } we write ℓ p 2 . The spaces ℓ p for p with 1 < p < ∞ are known to b e strictly conv ex [18, Exercise 16.201] and hence so are their dual spaces. Thus, in a certain sense the space ℓ 1 is a simple example of an infinite-dimensional computable normed space whose dual ℓ ∞ is not strictly con vex. Indeed the Hahn-Banac h Theorem already exhibits its full complexit y on ℓ 1 . One of our main results is the follo wing. Theorem 5. HBT ℓ 1 ≡ W WKL . Computabilit y of the Hahn-Banac h Theorem Revisited 3 W e are going to prov e this result in Section 5. In Section 6 we strengthen this result and we show that it do es not mak e the problem HBT ℓ 1 simpler, if w e pro vide more information on the subspace A using its distance function. Historically , Helly first prov ed the Hahn-Banach theorem for sp ecific normed spaces X inductively , starting from the following one-step version of the theo- rem [18, Theorem 7.3.1]. Theorem 6 (One-step Hahn-Banach theorem). L et X b e a norme d sp ac e with a line ar subsp ac e A ⊆ X and x ∈ X . Then every line ar functional f : A → R with ∥ f ∥ ≤ 1 has a line ar extension g : A + R x → R with ∥ g ∥ ≤ 1 . In fact, a line ar extension g : A + R x → R of f satisfies ∥ g ∥ ≤ 1 if and only if sup y ∈ A ( f ( y ) − ∥ x − y ∥ ) ≤ g ( x ) ≤ inf y ∈ A ( f ( y ) + ∥ x − y ∥ ) . In p articular, b oth b ounds exist and fal l into the interval [ −∥ x ∥ , ∥ x ∥ ] . Since computing a v alue in b et ween t w o n umbers that are given as a supre- m um or infim um, resp ectiv ely , is exactly what the intermediate v alue theorem en- ables us to do, we can conclude that the one-step Hahn-Banach theorem ( HBT 1 ) is reducible to the in termediate v alue theorem ( IVT ). Prop osition 7. HBT 1 X ≤ W IVT for every c omputable norme d sp ac e X . Our second main result that we pro ve in Section 3 is that also the one-step theorem exhibits its most complex b eha vior already on the space X = ℓ 1 . Theorem 8. HBT 1 ℓ 1 ≡ W IVT . The historical pro ofs of the Hahn-Banac h theorem, provided by Banach and Hahn, proceed from the one-step v ersion inductiv ely using transfinite induc- tion [18]. No wada ys, the text b o ok pro ofs use Zorn’s lemma instead and in both cases the pro ofs are based on the axiom of c hoice. In the case of separable spaces, it is well-kno wn [18] that the axiom of dep enden t choice is sufficient to obtain the Hahn-Banach theorem inductiv ely from the one-step version. W e can mimic this pro of and in this wa y we obtain a new proof of the upp er bound. Prop osition 9. HBT X ≤ W IVT ∞ for every c omputable norme d sp ac e X . Here IVT ∞ denotes the infinite pro duct of IVT in the sense of [5], where it is also prov ed that WKL ∞ ≡ W WKL and hence IVT ∞ ≡ W WKL follows. Kihara no- ticed that the infinite product operation corresponds to the axiom of dependent c hoice in some logical setting. 3 If the space X under consideration is finite-dimensional, then a finite num b er of loops is sufficien t to obtain the Hahn-Banac h theorem. In fact, we obtain the follo wing result. 3 T ak a yuki Kihara, The infinite lo op op er ation and the axiom of dep endent choic e , presen tation at CCA 2025, Kyoto, Japan, 24 Septem b er 2025. 4 V. Brattk a and C. Sorg Prop osition 10. HBT X ≤ W IVT [ n − 1] for every n –dimensional c omputable nor- me d sp ac e X and n ≥ 1 . Here IVT [ n ] denotes the n –fold comp ositional pro duct of IVT . W e note that IVT < W IVT [2] [10, Theorem 9.3]. The intermediate v alue theorem IVT is kno wn to hav e computable solutions for computable instances. This gives us imme- diately a non-uniform computable version of the Hahn-Banac h theorem as a corollary . Corollary 11 (Metakides and Nero de 1982). L et X b e a finite-dimensional c omputable norme d sp ac e with a c omputably sep ar able subsp ac e A ⊆ X . Then every c omputable line ar functional f : A → R has a c omputable line ar extension g : X → R with ∥ g ∥ = ∥ f ∥ . Finally , we note that the upp er b ound given in Prop osition 10 is not tight. In fact, w e prov e the following result in Section 4 for the t wo-dimensional case. Theorem 12. HBT ℓ 1 2 ≡ W LLPO . The diagram in Figure 1 shows the relev ant W eihrauc h degrees that we are going to study . HBT ℓ 1 2 ≡ W HBT ℓ ∞ 2 ≡ W LLPO HBT 1 ℓ 1 ≡ W IVT ≡ W CC [0 , 1] HBT ℓ 1 ≡ W WKL ≡ W IVT ∞ Fig. 1. The Hahn-Banac h theorem in the W eihrauch lattice. 2 W eihrauc h Complexit y and the Hahn-Banac h Theorem W e in tro duce concepts from computable analysis and W eihrauch complexit y that w e are going to use and w e refer the reader to [9,20] for more details. W e recall that a r epr esentation of a space X is a surjective partial map δ X : ⊆ N N → X . In this case ( X, δ X ) is called a r epr esente d sp ac e . If w e hav e tw o represen ted spaces ( X , δ X ) and ( Y , δ Y ) , then we automatically ha v e a represen tation δ C ( X,Y ) of the space of functions f : X → Y that hav e a contin uous realizer. A partial function F : ⊆ N N → N N is called a r e alizer of some partial multiv alued function f : ⊆ X ⇒ Y , if δ Y F ( p ) ∈ f δ X ( p ) for all p ∈ dom( f δ X ) . In this situation w e also write F ⊢ f . It is well-kno wn that there are universal functions U : ⊆ N N → N N suc h that for ev ery con tinuous Computabilit y of the Hahn-Banac h Theorem Revisited 5 F : ⊆ N N → N N there is some q ∈ N N suc h that F ( p ) = U ⟨ q , p ⟩ for all p ∈ dom( F ) . Here ⟨·⟩ denotes some standard pairing function on Baire space N N (w e use this notation for pairs as w ell as for the pairing of sequences). F or short we write U q ( p ) := U ⟨ q , p ⟩ for all q , p ∈ N N . Now we obtain a representation δ C ( X,Y ) of the set C ( X , Y ) of total singlev alued functions f : X → Y with contin uous realizers b y δ C ( X,Y ) ( q ) = f : ⇐ ⇒ U q ⊢ f . It is w ell-known that for admissibly represen ted T 0 –spaces X, Y the function space C ( X, Y ) consists exactly of the usual con tinuous functions (see [9,20] for more details). W e need a representation of function spaces C ( A, Y ) for v arying domains A . W e use copro ducts of the following form for this purpose. Definition 13 (Copro duct function spaces). Let X, Y b e represented spaces and let ( P ( X ) , δ P ) b e a represented space with P ( X ) ⊆ 2 X . Then C P ( X, Y ) := F A ∈P ( X ) C ( A, Y ) := { ( f , A ) : A ∈ P ( X ) and f ∈ C ( A, Y ) } denotes the c opr o duct function sp ac e that we represent b y δ C P , defined b y δ C P ⟨ q , p ⟩ = ( f , A ) : ⇐ ⇒ δ P ( p ) = A and U q ⊢ f for all total single-v alued functions f : A → Y in C ( A, Y ) with A ∈ P ( X ) . If Y = R , then w e write for short C P ( X ) := C P ( X, R ) . W e will use for P ( X ) for certain spaces of closed subsets of a computable metric space X . W e recall that a c omputable metric sp ac e X is a metric space ( X , d ) together with a dense sequence s : N → X suc h that d ◦ ( s × s ) is computable. By S ( X ) w e denote the space of non-empt y closed subsets A ⊆ X represented via a sequence ( x n ) n ∈ N suc h that A = { x n : n ∈ N } . By L ( X ) we denote the space of non-empty closed subsets A ⊆ X represen ted via their distanc e functions d A : X → R , x 7→ inf y ∈ A d ( x, y ) . In the first case the name of a set A is a name for a p oin t in X N , in the second case a name for a function in C ( X ) = C ( X , R ) . The computable sets A ∈ S ( X ) are called c omputably sep ar able and the computable sets A ∈ L ( X ) are called lo c ate d [11]. It follo ws from [11, Theorems 3.8, 3.9] that the representation of L ( X ) contains significantly more information on closed subspaces than that of S ( X ) . In particular, the iden tity id : L ( X ) → S ( X ) is computable for every computable metric space, but typically the in verse is not computable (not ev en for X = R ). W e write C S or C L instead of C P if w e use P ( X ) = S ( X ) or P ( X ) = L ( X ) , resp ectiv ely . W e recall that a c omputable norme d sp ac e X is a normed space ( X , ∥ · ∥ ) together with a fundamental se quenc e e : N → X (i.e., a se quence whose linear 6 V. Brattk a and C. Sorg span is dense) such that the induced metric space is a computable metric space. This metric space is defined via some standard enumeration s : N → X of the rational linear com binations of e . W e only consider normed spaces o ver the field of real n umbers R . It is w ell-known that linear maps on separable normed spaces can b e repre- sen ted by their v alues on a fundamental sequence together with a bound (see, e.g., [2, Theorem 4.3]). Prop osition 14 (Linear b ounded functionals). L et X b e a c omputable nor- me d sp ac e with fundamental se quenc e e : N → X . The multivalue d function F : ⊆ C ( X ) ⇒ R N × N , f 7→ { (( f ( e n )) n ∈ N , M ) : ∥ f ∥ ≤ M } , define d on al l line ar b ounde d functionals f , is c omputable and has a c omputable left inverse. W e can now define the Hahn-Banach pr oblem HBT X and the one-step Hahn- Banach pr oblem HBT 1 X . Definition 15 (Hahn-Banach problem). Let X b e a computable normed space. Then w e define: 1. HBT X : ⊆ C S ( X ) ⇒ C ( X ) , where HBT X ( f , A ) is the set { g | g : X → R is a linear extension of f with ∥ g ∥ ≤ ∥ f ∥} with dom( HBT X ) := { ( f , A ) | f : A → R linear with 0 < ∥ f ∥ < ∞} . 2. HBT 1 X : ⊆ C S ( X ) × X ⇒ C S ( X ) , where HBT 1 X ( f , A, x ) is the set { ( g , A + R x ) | g : A + R x → R is a linear extension of f with ∥ g ∥ ≤ ∥ f ∥} with dom( HBT 1 X ) := { ( f , A, x ) | f : A → R linear with 0 < ∥ f ∥ < ∞} . W e note that w e do not require x ∈ A for HBT 1 X . W e use the usual concept of W eihrauch reducibility (see [8] for a survey) in order to compare problems. In general a pr oblem is a m ultiv alued function f : ⊆ X ⇒ Y on represented spaces X, Y . Here id : N N → N N denotes the iden tity on Baire space. Definition 16 (W eihrauch reducibilit y). Let f : ⊆ X ⇒ Y and g : ⊆ Z ⇒ W b e problems. W e say that f is W eihr auch r e ducible to g , in symbols f ≤ W g , if there are computable H, K : ⊆ N N → N N suc h that H ⟨ id , GK ⟩ ⊢ f , whenever G ⊢ g holds. As usual, w e denote the corresp onding equiv alence b y ≡ W . W e will also need a num ber of w ell-known b enc hmark problems that we will use to charac- terize the ab o v e Hahn-Banac h problems. The problems in the follo wing defini- tion are known as W e ak Kőnig’s lemma ( WKL ), as sep ar ation pr oblem ( SEP ), as interme diate value the or em ( IVT ), as c onne cte d choic e pr oblem ( CC [0 , 1] ) of [0 , 1] and as lesser limite d principle of omniscienc e . W e write T r for the set of binary tr e es T ⊆ { 0 , 1 } ∗ represen ted via their c haracteristic functions. By range( f ) = { f ( x ) : x ∈ X } we denote the r ange of a function f : X → Y . Computabilit y of the Hahn-Banac h Theorem Revisited 7 Definition 17 (Benchmark problems). W e define the following problems: 1. WKL : ⊆ T r ⇒ 2 N , T 7→ [ T ] , where dom( WKL ) is the set of all infinite binary trees and [ T ] denotes the set of infinite paths of T . 2. SEP : ⊆ N N × N N ⇒ 2 N , ( p, q ) 7→ { A : ( ∀ n ) range( p ) ⊆ A ⊆ N \ range( q ) } , where dom( SEP ) := { ( p, q ) : range( p ) ∩ range( q ) = ∅ } . 3. IVT : ⊆ C [0 , 1] ⇒ [0 , 1] , f 7→ f − 1 { 0 } with dom( IVT ) = { f : f (0) · f (1) < 0 } . 4. CC [0 , 1] : ⊆ [0 , 1] N × [0 , 1] N ⇒ [0 , 1] , (( a n ) , ( b n )) 7→ { x : ( ∀ n ) a n ≤ x ≤ b n } with dom( CC [0 , 1] ) = { (( a n ) , ( b n )) : ( ∀ n ) a n ≤ a n +1 ≤ b n +1 ≤ b n } . 5. LLPO : ⊆ 2 N × 2 N ⇒ { 0 , 1 } , ( p 0 , p 1 ) 7→ { i ∈ { 0 , 1 } : p i = b 0 } with dom( LLPO ) := { ( p 0 , p 1 ) : ( ∃ i ∈ { 0 , 1 } ) p i = b 0 } . Here b 0 ∈ 2 N denotes the constant zero sequence. The following equiv alences are well-kno wn. The equiv alence of W eak K őnig’s lemma and the separation problem was pro ved by Gherardi and Marcone [14], the equiv alence of connected c hoice and the intermediate v alue theorem is due to Gherardi and the first au- thor [6, Prop osition 3.6, Theorem 6.2]. More results on connected c hoice can b e found in [10]. The pro of that W eak Kőnig’s lemma is closed under infinite loops can b e found in [5], as well as all other required results for infinite lo ops. Prop osition 18. WKL ∞ ≡ W WKL ≡ W SEP ≡ W IVT ∞ and IVT ≡ W CC [0 , 1] . W e now recall the definition of infinite lo ops , whic h were introduced in [5]. In tuitively , f ∞ = ... ⋆ f ⋆ f can b e seen as an infinite comp ositional pro duct. The c omp ositional pr o duct of tw o problems f , g on Baire space can b e defined b y f ⋆ g := ⟨ id × f ⟩ ◦ U ◦ ⟨ id × g ⟩ . By f [ n ] w e denote the n –fold comp ositional pro duct of f with itself. The infinite lo op is an in verse limit construction based on this pro duct. Here f ( A ) = S x ∈ A f ( x ) for f : ⊆ X ⇒ Y and A ⊆ X . Definition 19 (Infinite lo op). Let f : ⊆ N N ⇒ N N b e a problem. Then w e define the inverse limit f ∞ : ⊆ N N ⇒ N N of f by f ∞ ( q 0 ) := {⟨ q 0 , q 1 , q 2 , ... ⟩ ∈ N N : ( ∀ i ) q i +1 ∈ U ◦ ⟨ id × f ⟩ ( q i ) } where dom( f ∞ ) cons ists of all q 0 ∈ N N suc h that A 0 := { q 0 } ⊆ dom( U ◦ ⟨ id × f ⟩ ) and A i +1 := U ◦ ⟨ id × f ⟩ ( A i ) ⊆ dom( U ◦ ⟨ id × f ⟩ ) for all i ∈ N . This definition can b e extended from problems on Baire space to problems on arbitrary represen ted spaces using standard tec hniques. It has also b een prov ed in [5] that f 7→ f ∞ is a monotone op eration with resp ect to (strong) W eihrauc h reducibilit y . Now w e are well prepared to prov e our main results. 3 The One-Step Hahn-Banac h Theorem The wa y we hav e defined CC [0 , 1] mak es Prop osition 7 a direct corollary of Prop o- sition 18 and Theorem 6. That is, the one-step Hahn-Banach theorem is reducible to the in termediate v alue theorem for every computable normed space X . 8 V. Brattk a and C. Sorg Prop osition 20. HBT 1 X ≤ W CC [0 , 1] for every c omputable norme d sp ac e X . Pr o of. Giv en a functional f : A → R with 0 < ∥ f ∥ < ∞ and x ∈ X we can assume that ∥ f ∥ ≤ 1 , b ecause w e can just divide f by an upp er b ound M of ∥ f ∥ that can b e computed according to Prop osition 14. Then we obtain an extension g : A + R x → R with ∥ g ∥ ≤ 1 , where w e determine g ( x ) with the help of CC [0 , 1] and Theorem 6. By Prop osition 14 w e can actually compute g as a p oin t in the function space C ( A + R x ) with the av ailable information. In order to conv ert g into an extension of the original functional, we hav e to multiply it with M again. ⊓ ⊔ W e emphasize that this pro of only yields an ordinary W eihrauch reduction, not a strong one. If we apply the one-step version of the Hahn-Banac h prob- lem rep eatedly in an infinite lo op, then w e get a new pro of of the well-kno wn reduction to W eak K őnig’s lemma. Theorem 21. HBT X ≤ W WKL for every c omputable norme d sp ac e. Pr o of. Starting from a functional f : A → R for which we can again assume ∥ f ∥ ≤ 1 , we can just rep eatedly apply HBT 1 X for the fundamen tal sequence ( e n ) n of the space X . Inductiv ely , starting from f 0 := f and A 0 := A this yields functionals h n +1 : A n + R e n → R and closed sets A n +1 as closure of the linear span of A n + R e n . Using Prop osition 14 w e can compute extensions of each h n +1 to a functional of t yp e f n +1 : A n +1 → R that is used for the next application of HBT 1 X . Hence, every f n +1 is a linear extension of f n with ∥ f n ∥ ≤ 1 . Altogether, the v alues ( f n +1 ( e n )) n ∈ N determine a linear functional g : X → R that extends f with ∥ g ∥ ≤ 1 and these data suffice to obtain g as a p oin t in C ( X ) b y Proposition 14. Altogether, by Proposition 18, this prov es HBT X ≤ W IVT ∞ ≡ W WKL . ⊓ ⊔ In the finite-dimensional case, the same argument requires onl y finitely many applications of IVT , which yields Proposition 10. In the next section we will see that this b ound is not sharp, not even for the ℓ 1 –norm on R 2 . Next we wan t to pro ve that the one-step Hahn-Banach theorem reaches its maximal complexit y for X = ℓ 1 . W e use the computable linear isometry R : ℓ 1 ( N × { 0 , 1 } ) → ℓ 1 , R (( x n,i ) ( n,i ) ∈ N ×{ 0 , 1 } )(2 n + i ) := x n,i , (1) whic h allows us to identify ℓ 1 ( N × { 0 , 1 } ) with ℓ 1 . W e also use the standard fundamen tal sequence ( e n,i ) of unit v ectors of ℓ 1 ( N × { 0 , 1 } ) . Prop osition 22. CC [0 , 1] ≤ W HBT 1 ℓ 1 . Pr o of. Giv en tw o sequences ( a n ) n ∈ N and ( b n ) n ∈ N of rational num b ers in [0 , 1] suc h that a n ≤ a n +1 and b n +1 ≤ b n with a := sup n ∈ N a n ≤ inf n ∈ N b n =: b , the goal is to find a real num b er y ∈ [0 , 1] with a ≤ y ≤ b . Without loss of generalit y , w e can even assume a n < b n for all n ∈ N . W e work with the space X = ℓ 1 ( N × { 0 , 1 } ) that is isomorphic to ℓ 1 b y (1). W e now compute a functional f : A → R on a subspace A ⊆ X with ∥ f ∥ ≤ 1 Computabilit y of the Hahn-Banac h Theorem Revisited 9 and a p oin t x ∈ X such that ev ery linear extension g : A + R x → R of f with ∥ g ∥ ≤ 1 satisfies a ≤ g ( x ) ≤ b . This prov es CC [0 , 1] ≤ W HBT 1 ℓ 1 . In order to construct f , we first compute α n := a n − b n 2 < 0 and β n := a n + b n 2 ∈ [0 , 1] from the input data and then for all n ∈ N u n := e n, 0 + α n e n, 1 and v n := e n, 0 − e n +1 , 0 . Hence, we can also compute the closure A ∈ S ( X ) of the linear span of B := { v n , u n : n ∈ N } and x := P n ∈ N 2 − n − 1 e n, 0 ∈ X . Since B is linearly indep enden t, there is a unique linear f 0 : span( B ) → R with the v alues f 0 ( v n ) := 0 and f 0 ( u n ) := β n for all n ∈ N . W e claim that f 0 is b ounded with ∥ f 0 ∥ ≤ 1 and hence it extends uniquely to a linear b ounded functional f : A → R with ∥ f ∥ ≤ 1 by the Hahn- Banac h theorem. W e contin ue assuming this claim for the moment. Let h : A + R x → R b e a functional that we receive as output of HBT 1 X ( f , A, x ) . By the classical Hahn-Banach theorem h has an extension g : X → R that is a linear contin uous extension of f with ∥ g ∥ ≤ 1 . Since Y = ℓ ∞ ( N × { 0 , 1 } ) is the dual space of X , there is a w ∈ Y with ∥ w ∥ ∞ ≤ 1 such that g ( z ) = ⟨ w , z ⟩ := X k ∈ N ( w k, 0 z k, 0 + w k, 1 z k, 1 ) for all z ∈ X (here ⟨·⟩ simply denotes the dualit y pairing). Then 0 = g ( v n ) = ⟨ w , e n, 0 − e n +1 , 0 ⟩ = w n, 0 − w n +1 , 0 and hence the v alues y := w n, 0 are constan t for all n ∈ N . W e also obtain β n = g ( u n ) = ⟨ w , e n, 0 + α n e n, 1 ⟩ = w n, 0 + α n w n, 1 = y + α n w n, 1 for all n ∈ N . Since ∥ w ∥ ∞ ≤ 1 , we ha ve | w n, 1 | ≤ 1 and hence y = β n − α n w n, 1 ∈ [ β n + α n , β n − α n ] = [ a n , b n ] for all n ∈ N , which implies a ≤ y ≤ b . The preceding argument can also b e reversed. If we start with some arbitrary y with a ≤ y ≤ b , then we can c ho ose w n, 1 with | w n, 1 | ≤ 1 suc h that β n = y + α n w n, 1 and w n, 0 = y . Then w = ( w n, 0 , w n, 1 ) n ∈ Y is a p oin t with ∥ w ∥ ∞ ≤ 1 that hence defines a functional g : X → R , z 7→ ⟨ w , z ⟩ with ∥ g ∥ ≤ 1 and this functional extends f 0 b y the same calculation as ab o v e. This prov es the claim that f 0 can b e extended to f : A → R with ∥ f ∥ ≤ 1 . If g : X → R is now an extension of f with ∥ g ∥ ≤ 1 as ab o ve, then we can ev aluate g on x and w e obtain g ( x ) = * w , X n ∈ N 2 − n − 1 e n, 0 + = X n ∈ N 2 − n − 1 w n, 0 = X n ∈ N 2 − n − 1 y = y ∈ [ a, b ] . This implies h ( x ) = y ∈ [ a, b ] and completes the proof. □ 10 V. Brattk a and C. Sorg T ogether with Prop osition 20 we obtain the desired c haracterization. Corollary 23. HBT 1 ℓ 1 ≡ W IVT ≡ W CC [0 , 1] . 4 The Hahn-Banac h Theorem for ℓ 1 2 W e iden tify the space ℓ p 2 with R 2 equipp ed with the ℓ p –norm. In this section we w ant to prov e that for ℓ 1 2 (and ℓ ∞ 2 ), the Hahn-Banac h theorem is equiv alent to LLPO , which shows that the upp er b ound given in Prop osition 10 is not tigh t, not ev en for the ℓ 1 –norm. In fact, it suffices to consider the case of ℓ 1 , as there is a computable linear isometric map S : R 2 → R 2 , ( u, v ) 7→  u − v 2 , u + v 2  (2) that satisfies ∥ S ( u, v ) ∥ 1 = 1 2 ( | u + v | + | u − v | ) = max( | u | , | v | ) = ∥ ( u, v ) ∥ ∞ . In Figure 2 the resp ectiv e unit balls are illustrated. W e consider the case X = ℓ 1 2 . If we hav e a functional f : A → R with ∥ f ∥ = 1 defined on a one-dimensional subspace A ⊆ R 2 , then the extension of this functional to a functional g : R 2 → R with ∥ g ∥ = 1 is actually uniquely determined, provided that A do es not cross an y corner of the unit ball. This is b ecause f − 1 { 1 } is an affine subspace that is not allo wed to run through the interior of the ball (because ∥ f ∥ = 1 ) and hence it has to include one of the sides of the unit ball, which fixes all the v alues of the extension. One can use LLPO to determine on whic h side of the unit ball the affine h yp erplane f − 1 { 1 } lies. W e recall that LLPO is equiv alen t to the problem of determining one of the cases r ≤ 0 or r ≥ 0 , whic h holds for a real r ∈ R . F or the other direction of the reduction w e use an idea of Ishihara [15]. x y 0 ℓ 1 A x y 0 ℓ ∞ Fig. 2. Unit balls in R 2 with resp ect to ℓ 1 and ℓ ∞ . Prop osition 24. HBT ℓ 1 2 ≡ W HBT ℓ ∞ 2 ≡ W LLPO . Pr o of. W e consider the case of X = ℓ 1 2 , i.e., R 2 with the ℓ 1 –norm. Giv en a functional f : A → R for some linear subspace A ⊆ R 2 with 0 < ∥ f ∥ ≤ 1 . In Computabilit y of the Hahn-Banac h Theorem Revisited 11 fact, we can assume ∥ f ∥ = 1 , as w e can divide f b y its norm (as the operator norm is computable for finite-dimensional space s [2]). W e know that A  = { 0 } since ∥ f ∥ > 0 . Hence, w e can find some 0  = x = ( x 0 , x 1 ) ∈ A with ∥ x ∥ 1 = 1 and we can find some i ∈ { 0 , 1 } with x i  = 0 . Without loss of generality , w e assume x 0 > 0 and f ( x 0 , x 1 ) = 1 . If also x 1  = 0 , then the subspace A is not a one-dimensional subspace that crosses one of the corners of the unit ball and the extension of f is uniquely determined. In fact, if x 1 > 0 , then the norm-preserving extension g is uniquely determined by the additional condition g (0 , 1) = 1 , and if x 1 < 0 , then it is uniquely determined by g (0 , − 1) = 1 . If x 1 = 0 , then b oth of these norm-preserving extensions are p ossible. With the help of LLPO w e can select one of these cases. The remaining cases are handled analogously . Altogether, this sho ws HBT ℓ 1 2 ≤ W LLPO . F or the pro of of LLPO ≤ W HBT ℓ 1 2 w e follow a construction of Ishihara [15]. Giv en r ∈ R w e consider x = (1 , r ) ∈ R 2 , the subspace A := { ax : a ∈ R } ⊆ R 2 and the functional f : A → R with f ( ax ) := a · ∥ x ∥ 1 = a (1 + | r | ) . Then ∥ f ∥ = 1 . Let g : R 2 → R b e a linear extension of f with ∥ g ∥ = 1 . Then | g ( e i ) | ≤ 1 holds for the t wo unit v ectors e 1 = (1 , 0) , e 2 = (0 , 1) ∈ R 2 . Hence 1 + | r | = f ( x ) = g ( x ) = g ( e 1 + r e 2 ) = g ( e 1 ) + r g ( e 2 ) ≤ g ( e 1 ) + | r | . (3) Then g ( e 1 ) = 1 and hence r g ( e 2 ) = | r | . In order to chec k whether r ≤ 0 or r ≥ 0 w e just ha ve to find out whether g ( e 2 ) > − 1 or g ( e 2 ) < 1 . These conditions are semi-decidable in the input and can hence b e tested in parallel. Dep ending on whic h one is witnessed first, we output 1 or 0 , resp ectiv ely . Since r > 0 implies g ( e 2 ) = 1 and r < 0 implies g ( e 2 ) = − 1 , only one test can succeed in these cases. If r = 0 , b oth tests can succeed. The statemen t for ℓ ∞ 2 follo ws using the computable isometry (2) ⊓ ⊔ 5 The Hahn-Banac h Theorem for ℓ 1 F or their pro of of the reduction WKL ≤ W HBT Gherardi and Marcone [14] fol- lo wed the construction of Bro wn and Simpson [12,19], who in turn used ideas of similar constructions of Bishop [1], Metakides, Nerode and Shore [16,17]. W e briefly recall the construction due to Gherardi and Marcone. T o every instance ( p, q ) ∈ N N × N N of the separation problem, i.e., with range( p ) ∩ range( q ) = ∅ , they asso ciate a Banach space ( X p,q , ∥ · ∥ p,q ) that is defined as follo ws. Firstly , δ n :=      2 − k − 1 if k = min { i ∈ N : p ( i ) = n } exists − 2 − k − 1 if k = min { i ∈ N : q ( i ) = n } exists 0 otherwise . and then ε n := 1 − δ n 1+ δ n for all n ∈ N . Then one can obtain norms on R 2 b y ∥ ( α, β ) ∥ p,q ,n :=        max( | ε n α + β | , | α − β | ) if ε n < 1 , max( | α + β | , | ε − 1 n α − β | ) if ε n > 1 , max( | α + β | , | α − β | ) if ε n = 1 . 12 V. Brattk a and C. Sorg F or x = ( α n , β n ) n ∈ N ∈ ( R 2 ) N w e use the notation x n = ( α n , β n ) and x n, 0 = α n and x n, 1 = β n . No w one obtains a Banach space ( X p,q , ∥ · ∥ p,q ) with X p,q := n x ∈ ( R 2 ) N : ∥ x ∥ p,q < ∞ o , where ∥ x ∥ p,q := ∞ X n =0 2 − n − 1 ∥ x n ∥ p,q ,n . That is, ∥ · ∥ p,q is a ℓ 1 –sum of weigh ted ℓ ∞ –blo c ks in ( R 2 , ∥ · ∥ p,q ,n ) . On this Banac h space Gherardi and Marcone considered the functional f p,q : A p,q → R , x 7→ ∞ X n =0 2 − n − 1 x n, 0 (4) for the subspace A p,q := { x ∈ X p,q : x n, 1 = 0 for all n ∈ N } . (5) Then ∥ f p,q ∥ = 1 and from a functional g : X p,q → R that extends f p,q with ∥ g ∥ = 1 one can compute a set B ⊆ N that separates range( p ) and range( q ) , as n ∈ range( p ) = ⇒ g ( z n ) = − 2 − n − 1 and n ∈ range( q ) = ⇒ g ( z n ) = +2 − n − 1 (6) for all n ∈ N , where z n ∈ X p,q is defined b y ( z n ) n = (0 , 1) and ( z n ) k = (0 , 0) for k  = n . In order to prov e that the Hahn-Banach theorem exhibits its maximal pow er for ℓ 1 , w e construct a computable linear isometry . Prop osition 25. Ther e exists a line ar isometry F p,q : X p,q → ℓ 1 that is c om- putable uniformly in instanc es ( p, q ) ∈ N N × N N of the sep ar ation pr oblem. Pr o of. The linear map T n : R 2 → R 2 with T n ( α, β ) :=      ( ε n α + β , α − β ) if ε n < 1 ( α + β , ε − 1 n α − β ) if ε n > 1 ( α + β , α − β ) if ε n = 1 is computable uniformly in n relativ e to p, q and satisfies ∥ ( α, β ) ∥ p,q ,n = ∥ T n ( α, β ) ∥ ∞ . No w w e use the computable linear map S : R 2 → R 2 from (2) Hence, for S T n = S ◦ T n and x ∈ R 2 w e obtain | ( S T n ( x )) 0 | + | ( S T n ( x )) 1 | = ∥ S T n ( x ) ∥ 1 = ∥ x ∥ p,q ,n . No w we can define F p,q : X p,q → ℓ 1 b y F p,q ( x )(2 n + i ) := (2 − n − 1 S T n ( x n )) i for all x ∈ X p,q , n ∈ N and i ∈ { 0 , 1 } . Finally , w e obtain ∥ F p,q ( x ) ∥ 1 = ∞ X n =0 2 − n − 1 ∥ S T n ( x n ) ∥ 1 = ∞ X n =0 2 − n − 1 ∥ x n ∥ p,q ,n = ∥ x ∥ p,q and F p,q is computable uniformly in ( p, q ) . ⊓ ⊔ Computabilit y of the Hahn-Banac h Theorem Revisited 13 Since F p,q : X p,q → ℓ 1 is an injective computable linear map on computable Banac h spaces, it has a computable in verse F − 1 p,q : range( F p,q ) → X p,q b y the computable version of the Banach In verse Mapping Theorem [4, Corollary 5.3]. Ho wev er, this do es not automatically hold uniformly in p, q . But since F p,q is ev en an isometry , the op erator norm of the inv erse is 1 and hence we obtain uniformit y in p, q by [4, Theorem 5.9]. This allo ws us to obtain the following conclusion. Prop osition 26. SEP ≤ W HBT ℓ 1 Pr o of. Giv en an instance ( p, q ) ∈ N N × N N of the separation problem, we can compute the functional f p,q : A p,q → R from (4) and we obtain a functional f : A → R with A := F p,q ( A p,q ) by f := f p,q ◦ F − 1 p,q . Since everything is uniform in p, q , we can compute ( f , A ) ∈ C S ( ℓ 1 ) . No w we can apply HBT ℓ 1 in order to obtain a linear extension g : ℓ 1 → R of f with ∥ g ∥ ≤ ∥ f ∥ . Then g ′ := g ◦ F p,q is a linear functional g ′ : X p,q → R that extends f p,q . Because F p,q is an isometry we ha ve ∥ g ′ ∥ = ∥ f p,q ∥ = 1 . Hence we obtain the v alues g ′ ( z n ) = g ◦ F p,q ( z n ) from whic h we can determine a separating set B ∈ SEP ( p, q ) using (6). □ No w we obtain the follo wing result. Corollary 27. HBT ℓ 1 ≡ W WKL . 6 The Hahn-Banac h Theorem for Lo cated Subspaces The pro of of Prop osition 26 shifts the complexity from the space X p,q in to the functional f : A → R and the subspace A ⊆ ℓ 1 using the isometry F p,q . Hence, it is a relev an t question whether the complexity of the Hahn-Banach theorem on ℓ 1 can b e reduced by pro viding more information on the subspace A . W e will prov e that this is not the case, ev en if w e provide the subspace in form of its distance function d A , i.e., if S ( ℓ 1 ) is replaced by L ( ℓ 1 ) in the definition of HBT ℓ 1 . W e first pro ve that the subspaces from (5) can be computed as p oin ts in L ( X p,q ) . Prop osition 28. The sets A p,q ∈ L ( X p,q ) c an b e c ompute d uniformly in in- stanc es ( p, q ) ∈ N N × N N of the sep ar ation pr oblem. Pr o of. The definition of ∥ · ∥ p,q ,n implies that ∥ ( α, β ) ∥ p,q ,n ≥ | β | = ∥ (0 , β ) ∥ p,q ,n and ∥ ( α, β ) ∥ p,q ,n ≥ 1 2 | α | = 1 2 ∥ ( α, 0) ∥ p,q ,n for all ( α, β ) ∈ R 2 . In particular, ( α n , β n ) n ∈ X p,q implies ( α n , 0) n ∈ X p,q . Hence, the follo wing infimum for ( α n , β n ) n ∈ X p,q is attained for α ′ n = α n with the giv en v alue d A p,q (( α n , β n ) n ) = inf ( α ′ n , 0) n ∈ X p,q ∞ X n =0 2 − n − 1 ∥ ( α n − α ′ n , β n ) ∥ p,q ,n = ∞ X n =0 2 − n − 1 | β n | . Th us, d A p,q is computable in p, q , since P ∞ n =0 2 − n − 1 | β n | ≤ ∥ ( α n , β n ) ∥ p,q . ⊓ ⊔ Next w e prov e that lo catedness is preserved by computable linear isometries. 14 V. Brattk a and C. Sorg Prop osition 29. L et F : X → Y b e a c omputable bije ctive line ar isometry on c omputable norme d sp ac es ( X , ∥ · ∥ X ) and ( Y , ∥ · ∥ Y ) . Then F : L ( X ) → L ( Y ) , A 7→ F ( A ) is c omputable. This even holds uniformly in F . Pr o of. Since F is a bijectiv e linear isometry , we obtain d F ( A ) ( x ) = inf z ∈ F ( A ) ∥ x − z ∥ Y = inf y ∈ A ∥ F ( F − 1 ( x ) − y ) ∥ Y = inf y ∈ A ∥ F − 1 ( x ) − y ∥ X = d A ( F − 1 ( x )) for all x ∈ Y . If F is given, then w e can compute F − 1 b y [4, Theorem 5.9] as ∥ F − 1 ∥ = 1 . If, additionally , A ∈ L ( X ) is given in form of d A ∈ C ( X ) , then w e can compute d F ( A ) ∈ C ( Y ) by the equation abov e and hence F ( A ) ∈ L ( Y ) . □ No w we can transfer the proof of Prop osition 26 from S ( ℓ 1 ) to L ( ℓ 1 ) . Corollary 30. HBT ℓ 1 ≡ W WKL , even if the sp ac e S ( ℓ 1 ) in the definition of HBT ℓ 1 is r eplac e d by L ( ℓ 1 ) . The follo wing corollary strengthens Corollary 3. Corollary 31. Ther e exists a c omputable line ar functional f : A → R on a lo c ate d close d subsp ac e A ⊆ ℓ 1 with ∥ f ∥ = 1 and without a c omputable line ar extension g : ℓ 1 → R with ∥ g ∥ = 1 . A ckno wledgmen ts. W e ac knowledge funding by the German Researc h F oundation (DF G, Deutsche F orsch ungsgemeinschaft) – pro ject n umber 554999067 and b y the National Research F oundation of South Africa (NRF) – grant num b er 151597. References 1. 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