International Journal of Analysis and Applications ISSN 2291-8639 Volume 11, Number 2 (2016), 93-100 http://www.etamaths.com COMPUTABLE FRAMES IN COMPUTABLE BANACH SPACES S.K. KAUSHIK1,∗ AND POONAM MANTRY2 Abstract. We develop some parts of the frame theory in Banach spaces from the point of view of Computable Analysis. We define computable M-basis and use it to construct a computable Banach space of scalar valued sequences. Computable Xd frames and computable Banach frames are also defined and computable versions of sufficient conditions for their existence are obtained. 1. Introduction In functional analysis, a sequence (xn) in a Banach space X is called a M-basis, if it is complete in X and there exists a total sequence of functionals (fn) ⊆ X∗ such that (xn,fn) is a biorthogonal system. Not every separable Banach space has a Schauder basis but it has at least a bounded and norming M-basis. A Banach space with a M-basis is linearly isometric to the associated Banach space Xd = {(fn(x)) : x ∈ X} by a result in [17]. The notion of computable Banach spaces with computable basis has already been discussed in [7]. We define computable M-basis and prove the computable version of above result in the framework of computable analysis. The above mentioned computable Banach space Xd is generalized to a computable BK-space, which is further used to define computable Xd frame. A sufficient condition for the existence of a computable Xd frame is obtained. Computable version of a necessary and sufficient condition for the existence of Xd Bessel sequence [9], is also obtained. Finally, we define the concept of computable Banach frame and obtain some sufficient conditions for their existence. 2. Computable Analysis In this section, we briefly summarize some notions from computable analysis as presented in [19]. Computable Analysis is the Turing machine based approach to computability in analysis. Pioneering work in this field has been done by Turing [18], Grzegorczyk [11], Lacombe [13], Banach and Mazur [1], Pour-El and Richards [16], Kreitz and Weihrauch [12] and many others. The basic idea of the representation based approach to computable analysis is to represent infinite objects like real numbers, functions or sets, by infinite strings over some alphabet Σ (which at least contains the symbols 0 and 1). Thus, a representation of a set X is a surjective function δ :⊆ Σω → X where Σω denotes the set of infinite sequences over Σ and the inclusion symbol indicates that the mapping might be partial. Here, (X,δ) is called a represented space. Between two represented spaces, we define the notion of a computable function. Definition 2.1. [6] Let (X,δ) , (Y,δ′) be represented spaces. A function f :⊆ X → Y is called (δ,δ′) -computable if there exists a computable function F :⊆ Σω → Σω such that δ′F(p) = fδ(p) for all p ∈ dom(fδ) . A function F :⊆ Σω → Σω is said to be computable if there exists some Turing machine, which computes infinitely long and transforms each sequence p, written on the input tape, into the corresponding sequence F(p) , written on one way output tape. We simply call, a function f computable, if the represented spaces are clear from the context. For comparing two representations,δ, δ ′ of a set X , we have the notion of reducibility of representations. δ is called reducible to δ ′ ,δ ≤ δ ′ (in symbols), if there exists a computable function 2010 Mathematics Subject Classification. 03F60, 46S30. Key words and phrases. computable function; computable Banach space. c©2016 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 93 94 KAUSHIK AND MANTRY F :⊆ Σω → Σω such that δ(p) = δ ′ F(p) for all p ∈ dom(δ) . This is equivalent to the fact that the identity I : X → X is (δ,δ ′ ) - computable. If δ ≤ δ ′ and δ ′ ≤ δ, then δ and δ ′ are called computably equivalent. Analogously to the notion of computability ,we can define the notion of (δ,δ ′ ) -continuity, by substituting a continuous function F in the definitions above. On Σω , we use the Cantor topology, which is simply the product topology of the discrete topology on Σ . Given a represented space (X,δ) , a computable sequence is defined as a computable function f : N → X where we assume that N is represented by δN(1n0ω) = n and a point x ∈ X is called computable if there is a constant computable sequence with value x. The notion of (δ,δ ′ ) -continuity agrees with the ordinary topological notion of continuity, as long as, we are dealing with admissible representations. A representation δ of a topological space X is called admissible if δ is continuous and if the identity I : X → X is (δ′,δ) -continuous for any continuous representation δ′ of X . If δ, δ′ , are admissible representation of topological spaces X ,Y , then a function f :⊆ X → Y is (δ,δ′) continuous iff it is sequentially continuous [5]. Given two represented spaces (X,δ) , (Y,δ′) , there is a canonical representation [δ → δ′] of the set of (δ,δ′) - continuous functions f : X → Y . If δ and δ′ are admissible representations of sequential topological spaces X and Y respectively, then [δ → δ′] is actually a representation of the set C(X,Y ) of continuous functions f : X → Y . The function space representation can be characterized by the fact that it admits evaluation and type conversion. See [19] for details. If (X,δ) , (Y,δ′) are admissibly represented sequential topological spaces, then, in the following, we will always assume that C(X,Y ) is represented by [δ → δ′] . It follows by evaluation and Type conversion that the computable points in (C(X,Y ), [δ → δ′]) are just the (δ,δ′) -computable functions f :⊆ X → Y [19]. For a represented space (X,δ) , we assume that the set of sequences XN is represented by δN = [δN → δ] . The computable points in (XN,δN) are just the computable sequences in (X,δ) . The notion of computable metric space was introduced by Lacombe [14]. However, we state the following definition given by Brattka [6]. Definition 2.2. ([6]) A tuple (X,d,α) is called a computable metric space if (1) (X,d) is a metric space. (2) α : N → X is a sequence which is dense in X . (3) do(α×α) : N2 → R is a computable (double) sequence in R. Given a computable metric space (X,d,α) , its Cauchy Representation δX :⊆ Σω → X is defined as δX (01 n0+101n1+101n2+1...) := lim i→∞ α(ni) for all ni ∈ N such that (α(ni))i∈N converges and d(α(ni),α(nj)) < 2 −i for all j > i. In the following, we assume that computable metric spaces are represented by their Cauchy representation. All Cauchy representations are admissible with respect to the corresponding metric topology. An Example of a computable metric space is (R,dR,αR) with the Euclidean metric dR(x,y) = ‖x−y‖ and a standard numbering of a dense subset Q ⊆ R as αR < i,j,k >= (i − j)/(k + 1) . Here, the bijective Cantor pairing function <,>: N2 → N is defined as < i,j >= j + (i + j)(i + j + 1)/2 and this definition can be extended inductively to finite tuples. It is known that the Cantor pairing function and the projections of its inverse are computable. In the following, we assume that R is endowed with the Cauchy representation δR induced by the computable metric space given above. Brattka [6] gave the following definition of a computable normed linear space. COMPUTABLE FRAMES IN COMPUTABLE BANACH SPACES 95 Definition 2.3. ([6]) A space (X,‖.‖,e) is called a computable normed space if:- (1) ‖.‖ : X → R is a norm on X . (2) The linear span of e : N → X is dense in X . (3) (X,d,αe) with d(x,y) =‖ x−y ‖ and αe < k,< n0, ...,nk >>= Σ k i=0αF (ni)ei is a computable metric space with Cauchy representation δX . It was observed that computable normed space is automatically a computable vector space, that is, the linear operations are all computable. If the underlying space (X,‖.‖) is a Banach space then (X,‖.‖,e) is called a computable Banach space. We always assume that computable normed spaces are represented by their Cauchy representations, which are admissible with respect to norm topology. Two computable Banach space with the same underlying set are called computably equivalent if the corresponding Cauchy representations are computably equivalent. A sequence (ei)i∈N in a Banach space X , is called a Schauder basis of X if every x ∈ X can be uniquely represented as x = Σ∞i=0xiei with xi ∈ F . If X is a computable Banach space, then a sequence (ei)i∈N is called a computable basis, if it is a Schauder basis of X that is computable in X . A sequence space S is called a BK space if it is a Banach space and the coordinate functionals are continuous on S . Here, a sequence space is a set S of sequences of scalars which is closed under co-ordinatewise addition and scalar multiplication. Several representations for the operator space B(X,Y ) of bounded linear operators between two computable normed spaces are defined in [3]. In the following, we state some of the representations for the operator space B(X,Y ) , as they are used in consequent results of this paper. Definition 2.4. ([3]) Let (X,‖.‖,e) and Y be computable normed spaces. Define representations of B(X,Y ) : 1) δev(p) = T ⇔ [δX → δY ](p) = T 2) δseq(p) = T ⇔ δNY (p) = (Tei)i∈N 3) δ≥seq < p,q >= T ⇔ δseq(p) = T and δR(q) ≥‖T‖. 3. Main Results In order to develop a systematic computable frame theory on Banach spaces, we first extend the notion of computability to M-basis. M-basis were introduced by A.I. Markusevic, who regarded them as a natural generalization of the trigonometric system in C[0, 2π] and hence, as a natural replacement for basis. M-basis exists in every separable Banach space. We begin with the following definition of computable M-basis. Definition 3.1. A sequence (xn) in a computable Banach space (X,‖.‖,e) is a Computable M-Basis of X if :- (1) (xn) is a computable complete sequence in X . (2) There exists a total computable sequence of functionals (fn) ⊆ X∗ , with respect to [δX → δF ] representation, such that (xn,fn) is a biorthogonal system. Let (X,‖.‖,e) be a computable Banach space with a computable M-basis (xn) . Since (xn) is a computable complete sequence in X , (X,‖.‖, (xn)) is a computable Banach space that is computably equivalent to (X,‖.‖,e) . Remark 3.2. Let (X,‖.‖, (en)) be a computable Banach space with computable basis (en) , then (en) is a computable complete sequence in X . The sequence (e ′ n) of co-ordinate functionals is a computable sequence in C(X,F) by Proposition 3.3 in [7]. Also, (e′n) is a total sequence of functionals such that (en,e ′ n) is a biorthogonal system. Hence, a computable basis in a computable Banach space is a computable M-Basis. 96 KAUSHIK AND MANTRY Next, we prove that an associated Banach space of scalar valued sequences with respect to a computable M-basis is a computable Banach space. Theorem 3.3. Let (xn) be a computable M-basis for a computable Banach space (X,‖.‖, (xn)) , with associated sequence of functionals (fn) . Let Ed = {(fn(x)) : x ∈ X} be the associated Banach space with norm ‖(fn(x))‖Ed = ‖x‖X , x ∈ X . Then (Ed,‖.‖, (ei)) , (ei) being the sequence of canonical unit vectors, forms a computable Banach space. Proof: Let d and d′ be the metric induced by the norm of Ed and X , respectively. Note that, for h =< k,< n0, ...nk >> and h ′ =< p,< m0, ...mp >> ∈ N, d(αe(h),αe(h ′)) = ‖Σki=0αF (ni)ei − Σ p i=0αF (mi)ei‖ = ‖(fn(Σki=0αF (ni)xi − Σ p i=0αF (mi)xi))‖Ed = ‖Σki=0αF (ni)xi − Σ p i=0αF (mi)xi‖X = d′(αx(h),αx(h ′)) Now, the result follows from the fact that (X,d′,αx) is a computable metric space. In the following result, we prove that the computable Banach space (X,‖.‖, (xn)) with computable M-basis (xn) and the associated computable Banach space (Ed,‖.‖, (ei)) are computably isomorphic. Here, a computable isomorphism T is an isomorphism such that T as well as T−1 are computable. Theorem 3.4. Let (X,‖.‖, (xn)) be a computable Banach space, (xn) be a computable M-basis and (Ed,‖.‖, (ei)) be the associated computable Banach space. Then the mapping T : X → Ed given by T(x) = (fn(x)) ,x ∈ X is a computable isometrical isomorphism. Proof: The map T is a bounded linear operator satisfying ‖Tx‖ = ‖x‖, for all x ∈ X and therefore, ‖T‖≤ 1 . Also, (T(xn)) = (en) is a computable sequence in Ed . Thus, we can get a δ≥seq name of T and so a δev name. Therefore, T is [δX → δEd ] computable. Hence, by computable Banach Inverse Mapping Theorem in [6], T is a computable isometrical isomorphism. Next, we generalize the notion of associated computable Banach space of scalar valued sequences. Definition 3.5. A BK space Xd is said to be a computable BK-space if it is a computable Banach space such that the sequence of co-ordinate functionals τj : Xd → F given by τj((xi)) = xj , j ∈ N is computable with respect to [δXd → δF ] representation. Example 3.6. Let (xn) be a computable M-basis for a computable Banach space (X,‖.‖, (xn)) , with associated total sequence of functionals (fn) ⊆ X∗ . Then Ed = {(fn(x)) : x ∈ X}, the associated Banach space is a BK space. Also, (Ed,‖.‖, (ei)) is a computable Banach space as proved above. The sequence of co-ordinate functionals τj : Ed → F given by τj((fn(x))) = fj(x),x ∈ X is computable in C(Ed,F) as ‖τn‖≤‖fn‖ and (fn) is a computable sequence with respect to [δX → δF ] representation. Also, τn(ek) = δkn , that is, given n and k, τn(ek) can be computed. Hence, (Ed,‖.‖, (ei)) is a computable BK-space. We define computable Xd frame for a computable Banach space in the following. Definition 3.7. Let X be a computable Banach space, Xd be a computable BK-space. A computable sequence (gi) ⊆ X∗ , with respect to [δX → δF ] representation is called a computable Xd frame for X if: (1) (gi(f)) ∈ Xd ∀f ∈ X. (2) ‖f‖X and ‖(gi(f))‖Xd are equivalent, that is, there exists constants A, B > 0 such that A‖f‖X ≤‖(gi(f))‖Xd ≤ B‖f‖X for all f ∈ X . If, only the (1) and the upper condition in (2) are satisfied, (gi) is called a computable Xd Bessel sequence for X . COMPUTABLE FRAMES IN COMPUTABLE BANACH SPACES 97 Example 3.8. Let (xn) ⊆ X , (fn) ⊆ X∗ and Xd = Ed be as in Example 3.6. Then (Xd,‖.‖, (ei)) is a computable BK space. Also, since (fn) ⊆ X∗ is a computable sequence with respect to [δX → δF ] representation such that (fn(x)) ∈ Xd for all x ∈ X and ‖(fn(x))‖Xd = ‖x‖X , x ∈ X , (fn) forms a computable Xd frame for X . In the following, we present a computable version of a sufficient condition for the existence of a Xd frame (Theorem 2.1 in [9]). Theorem 3.9. Let X be any computable Banach space and Xd be any computable BK space. If X is isometrically isomorphic to a subspace of Xd by a computable map, then there exists a computable Xd frame for X . Proof: Let T : X → Xd be a computable map such that X is isometrically isomorphic to T(X) . Let (τi) be the computable sequence of co-ordinate functionals of Xd , given by τi((xj)) = xi, i ∈ N, (xj) ∈ Xd . For each i ∈ N, define gi = τi ◦T . Then (gi) is a computable sequence of functionals such that (gi(f)) = (τi(T(f))) = Tf ∈ Xd ,f ∈ X and ‖(gi(f))‖Xd = ‖Tf‖Xd = ‖f‖X . Hence, (gi) ⊆ X ∗ is a computable Xd frame for X . For the converse, Theorem 2.1 in [9] states that if (gi) be an Xd frame for a Banach space X , then the map U : X → Xd given by U(f) = (gi(f)) is an isomorphism of X into Xd . Using the techniques from [8], we show that the computable version of this result does not hold. Example 3.10. Consider the computable Banach space (l2,‖.‖, (ei)) with the sequence of canonical unit vectors (ei) as computable basis. Let (ai) be a computable sequence of positive reals such that ‖(ai)‖l2 exists but is not computable. We assume a0 = 1 . Define a linear bounded operator T : l2 → l2 as   1 a1 a2 a3 · · · 0 1 0 0 · · · 0 0 1 0 · · · ... ... ... . . .   Then, (fi) = (Tei) is a frame for l2 . Define gi : l2 → R by gi(f) =< f,fi >, i ∈ N. Then, (gi) forms a computable Xd frame for l2 where Xd is the computable BK space (l2,‖.‖, (ei)) . But the operator U : l2 → l2 , U(f) = (< f,fi >) is not computable as U(e0) = (ai) is not computable in l2 . The next result shows that the map U is computable with respect to [δX → δNF ] representation. Theorem 3.11. Let X be a computable Banach space and Xd be a computable BK space. If (gi) ⊆ X∗ be a computable Xd frame for X then the map U : X → Xd f → (gi(f)) is (δX, [δN → δF ]) computable isomorphism of X into Xd . Proof: The map U is an isomorphism of X into Xd by Theorem 2.1 in [9]. By the computability of the sequence (gi) and the Evaluation property , we get that the map U ′ : N×X → F (i,f) → gi(f) is computable with respect to ([δN,δX ],δF ) representation. By Type Conversion, the map U is computable with respect to (δX, [δN → δF ]) representation. 98 KAUSHIK AND MANTRY Now, we give a computable version of a necessary and sufficient condition for the existence of a Xd Bessel sequence (Corollary 3.3 in [9]). First, we recall, the Dual space representation δX∗ of the dual space X ∗ as given in [7]. Definition 3.12. ([7]) For a separable Banach space X , define a representation δX∗ of the dual space X∗ by δX∗ < p,q >= f ⇔ [δX → δF ](p) = f and δR(q) = ‖f‖ . Next, we give the definition of computable dual space given in [7], as it is required in the subsequent result. Definition 3.13. ([7]) Let X be a computable Banach space. X is said to have a computable dual space X∗ if there exists a sequence e∗ : N → X∗ such that: (1) (X∗,‖.‖,e∗) is a computable Banach space. (2) δX∗ is computably equivalent to the Cauchy representation δ c X∗ of (X ∗,‖.‖,e∗) . Theorem 3.14. Let X be a computable Banach space with computable dual space. Let (Xd,‖.‖, (ei)) be a computable BK-space and (X∗d,‖.‖, (Ei)) be a computable Banach space, (ei) and (Ei) being the sequences of standard unit vectors as basis.Then if, (gi) ⊆ X∗ be a computable Xd -Bessel sequence for X with bound B with (‖gi‖) being a computable sequence then T : X∗d → X ∗ given by T((di)) = Σdigi is a well defined computable operator from ‖T‖≤ B . Converse holds if Xd is reflexive space. Proof: Define R : X → Xd by R(x) = (gi(x)) ,x ∈ X . Since (gi) is an Xd - Bessel sequence, (gi(x)) ∈ Xd , x ∈ X and ‖R(x)‖ = ‖(gi(x))‖Xd ≤ B‖x‖X , x ∈ X . Consider R ∗ : X∗d → X ∗ . Then R∗(Ej) : X → F is such that R∗(Ej)(x) = Ej(R(x)) = gj(x),x ∈ X . Therefore,R∗(Ej) = gj , for all j ∈ N such that R∗(ΣidiEi) = Σidigi . Thus, T = R∗ is a well defined operator given by T((di)) = Σidigi satisfying ‖T‖ ≤ B . As, (gi) ⊆ X∗ is a computable sequence with respect to [δX → δF ] representation and (‖gi‖) is assumed to be a computable sequence, therefore, (gi) is a computable sequence in X ∗ with respect to Cauchy representation. Therefore, T is [δcX∗ d → δcX∗] computable. Conversely, let T : X∗d → X ∗ be a computable operator given by T((di)) = Σidigi . Then T(Ei) = gi , for all i and since, a computable operator maps computable sequences to computable sequences, therefore, (gi) is a computable sequence in X ∗ with respect to [δX → δF ] representation and (‖gi‖) is a computable sequence. Consider T∗ : X∗∗ → X∗∗d which satisfies T ∗(f)(Ej) = f(gj) , for all f ∈ X∗∗ . Therefore, (gi(x)) = (T∗(i(x))(Ei)) ∈ X∗∗d , identified with T ∗(i(x)) , where i is the natural embedding of X into X∗∗ . Since Xd is reflexive, therefore, (gi(x)) ∈ Xd and satisfies ‖(gi(x))‖Xd = ‖T ∗(i(x))‖ = ‖T‖‖x‖≤ B‖x‖X Hence, (gi) ⊆ X∗ is a computable Xd Bessel sequence for X with bound B and computable sequence of norms. Banach frames were introduced by Grochenig [10] as a generalization of the notion of frames in Hilbert spaces. In the following definition, we introduce the notion of computable Banach frame. Definition 3.15. Let X be a computable Banach space, Xd be a computable BK-space. Given a computable linear operator S : Xd → X and a computable Xd frame (gi) ⊆ X∗ , we say that ((gi),S) is a computable Banach frame for X with respect to Xd if S((gi(x))) = x for all x ∈ X . Example 3.16. Let (xn) be a computable M-Basis and Xd = {(fn(x)) : x ∈ X} be a computable BK-space and (fn) ⊆ X∗ is a computable Xd frame. By Theorem 3.4, the map T : X → Xd given by x → (fn(x)) , x ∈ X is a computable isometrical isomorphism. Thus, ((fi),T−1) is a computable Banach frame for X with respect to Xd . COMPUTABLE FRAMES IN COMPUTABLE BANACH SPACES 99 Next, we give a sufficient condition for the existence of a computable Banach frame. In the following result, we call a closed subspace of a computable Banach space to be computably complemented if it is the range of a computable linear projection in the space. Clearly, a computably complemented subspace is a computable subspace as defined in [7]. Theorem 3.17. A computable Banach space X has a computable Banach frame with respect to a given computable BK space Xd if X is isometrically isomorphic to a computably complemented subspace of Xd by a computable map. Proof: Suppose T : X → Xd be a computable map which is an isometric isomorphism of X into Xd and T(X) be the computably complemented subspace of Xd . Then, there exists a computable projection P : Xd → Xd such that P(Xd) = T(X) and P2 = P . Define S : Xd → X by Sx = T−1Px, x ∈ Xd . Then, by computable Banach Inverse Mapping theorem, S is a computable linear operator. Let (τj) be the computable sequence of co-ordinate functionals of Xd and gi = T ∗(τi) , for all i ∈ N. Then, for x ∈ X , we have gi(x) = T∗τi(x) = τiT(x) , for all i ∈ N. Therefore, gi = τiT and hence, (gi) is a computable sequence in X ∗ with respect to [δX → δF ] representation. Also, Tx = (gi(x)) ∈ Xd , for all x ∈ X and ‖(gi(x))‖Xd = ‖Tx‖Xd = ‖x‖X . Also S((gi(x))) = T−1P((gi(x))) = T −1((gi(x))) = x. Thus, ((gi),S) is a computable Banach frame for X with respect to Xd . Finally, we give sufficient condition under which a computable Xd frame for X is a computable Banach frame for X . Theorem 3.18. Let X be a computable Banach space, (Xd,‖.‖, (ei)) be a computable BK space with sequence (ei) of canonical unit vectors. Let (gi) ⊆ X∗ be a computable Xd frame for X . If there exists a computable sequence (fi) ⊆ X such that Σicifi is convergent for all (ci) ∈ Xd and f = Σigi(f)fi , for all f ∈ X . Then, there exists a computable linear operator T : Xd → X such that ((gi),T) is a computable Banach frame for X with respect to Xd . Proof: Define Tn : Xd → X as Tn((ci)) = Σni cifi . 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