@ Applied General Topology c© Universidad Politécnica de Valencia Volume 4, No. 1, 2003 pp. 115–131 Effective representations of the space of linear bounded operators Vasco Brattka ∗ Abstract. Representations of topological spaces by infinite se- quences of symbols are used in computable analysis to describe compu- tations in topological spaces with the help of Turing machines. From the computer science point of view such representations can be consid- ered as data structures of topological spaces. Formally, a representation of a topological space is a surjective mapping from Cantor space onto the corresponding space. Typically, one is interested in admissible, i.e. topologically well-behaved representations which are continuous and characterized by a certain maximality condition. We discuss a number of representations of the space of linear bounded operators on a Banach space. Since the operator norm topology of the operator space is non- separable in typical cases, the operator space cannot be represented admissibly with respect to this topology. However, other topologies, like the compact open topology and the Fell topology (on the operator graph) give rise to a number of promising representations of operator spaces which can partially replace the operator norm topology. These representations reflect the information which is included in certain data structures for operators, such as programs or enumerations of graphs. We investigate the sublattice of these representations with respect to continuous and computable reducibility. Certain additional conditions, such as finite dimensionality, let some classes of representations col- lapse, and thus, change the corresponding graph. Altogether, a precise picture of possible data structures for operator spaces and their mutual relation can be drawn. 2000 AMS Classification: 03F60, 03D45, 26E40, 46S30, 68Q05. Keywords: Computable functional analysis, effective representations. ∗Work partially supported by DFG Grant BR 1807/4-1 116 Vasco Brattka 1. Preliminaries from Computable Analysis In this paper we will study representations of the set of linear bounded operators on Banach spaces from the point of view of computable analysis, which is the Turing machine based theory of computability on real numbers and other topological spaces. Pioneering work on this theory has been presented by Turing [23], Banach and Mazur [1], Lacombe [14] and Grzegorczyk [10]. Recent monographs have been published by Pour-El and Richards [18], Ko [11] and Weihrauch [26]. Certain aspects of computable functional analysis have already been studied by several authors, see for instance [16, 9, 24, 27, 28]. In this section we briefly summarize some notions from computable analysis. For details the reader is referred to [26]. 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 should at least contain the symbols 0 and 1). Thus, a representation of a set X is a surjective mapping δ :⊆ Σω → X and in this situation we will call (X,δ) a represented space. Here Σω denotes the set of infinite sequences over Σ and the inclusion symbol is used to indicate that the mapping might be partial. If we have two represented spaces, then we can define the notion of a computable function. Definition 1.1 (Computable function). Let (X,δ) and (Y,δ′) be represented spaces. A function f :⊆ X → Y is called (δ,δ′)–computable, if there exists some computable function F :⊆ Σω → Σω such that δ′F(p) = fδ(p) for all p ∈ dom(fδ). Of course, we have to define computability of functions F :⊆ Σω → Σω to make this definition complete, but this can be done via Turing machines: F is computable if there exists some Turing machine, which computes infinitely long and transforms each sequence p, written on the input tape, into the cor- responding sequence F(p), written on the one-way output tape. Later on, we will also need computable multi-valued operations f :⊆ X ⇒ Y , which are defined analogously to computable functions by substituting δ′F(p) ∈ fδ(p) for the equation in Definition 1.1 above. If the represented spaces are fixed or clear from the context, then we will simply call a function or operation f computable. For the comparison of representations it will be useful to have the notion of reducibility of representations. If δ,δ′ are both representations of a set X, then δ is called reducible to δ′, δ ≤ δ′ in symbols, if there exists a computable function F :⊆ Σω → Σω such that δ(p) = δ′F(p) for all p ∈ dom(δ). Obviously, δ ≤ δ′ holds, if and only if the identity id : X → X is (δ,δ′)–computable. Moreover, δ and δ′ are called equivalent, δ ≡ δ′ in symbols, if δ ≤ δ′ and δ′ ≤ δ. Analogously to the notion of computability we can define the notion of (δ,δ′)–continuity for single and multi-valued operations, by substituting a con- tinuous function F :⊆ Σω → Σω for the computable function F in the defini- tions above. On Σω we use the Cantor topology, which is simply the product Effective representations 117 topology of the discrete topology on Σ. The corresponding reducibility will be called continuous reducibility and we will use the symbols ≤t and ≡t in this case. Again we will simply say that the corresponding function is continuous, if the representations are fixed or clear from the context. If not mentioned otherwise, we will always assume that a represented space is endowed with the final topology induced by its representation. This will lead to no confusion with the ordinary topological notion of con- tinuity, as long as we are dealing with admissible representations. A represen- tation δ of a topological space X is called admissible, if δ is maximal among all continuous representations δ′ of X, i.e. if δ′ ≤t δ holds for all continuous representations δ′ of X. If δ,δ′ are admissible representations of T0–spaces X, Y with countable bases, then a function f :⊆ X → Y is (δ,δ′)–continuous, if and only if it is continuous in the ordinary topological sense. For an extension of these notions to larger classes of spaces cf. [20, 21]. Given a represented space (X,δ), we will occasionally use the notions of a computable sequence and a computable point. A computable sequence is a computable function f : N → X, where we assume that N = {0, 1, 2, . . .} is represented by δN(1n0ω) := n and a point x ∈ X is called computable, if there is a constant computable function with value x. Given two represented spaces (X,δ) and (Y,δ′), there is a canonical repre- sentation [δ,δ′] of X × Y and a representation [δ → δ′] of certain functions f : X → Y . If δ,δ′ are admissible representations of T0–spaces with countable bases, then [δ → δ′] is actually a representation of the set C(X,Y ) of continuous functions f : X → Y . If Y = R, then we write for short C(X) := C(X,R). The function space representation can be characterized by the fact that it admits evaluation and type conversion. Proposition 1.2 (Evaluation and type conversion). Let (X,δ), (Y,δ′) be ad- missibly represented T0–spaces with countable bases and let (Z,δ′′) be a repre- sented space. Then: (1) (Evaluation) ev : C(X,Y )×X → Y, (f,x) 7→ f(x) is ([[δ → δ′],δ],δ′)– computable, (2) (Type conversion) f : Z ×X → Y , is ([δ′′,δ],δ′)–computable, if and only if the function f̌ : Z → C(X,Y ), defined by f̌(z)(x) := f(z,x) is (δ′′, [δ → δ′])–computable. The proof of this proposition is based on a version of the smn– and utm– Theorems, and can be found in [26]. If (X,δ), (Y,δ′) are admissibly represented T0–spaces with countable bases, then in the following we will always assume that C(X,Y ) is represented by [δ → δ′]. It is known that the computable points in (C(X,Y ), [δ → δ′]) are just the (δ,δ′)–computable functions f : X → Y [26]. If (X,δ) is a represented space, then we will always 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). Moreover, we assume 118 Vasco Brattka that Xn is always represented by δn, which can be defined inductively by δ1 := δ and δn+1 := [δn,δ]. To make this paper as self-contained as possible, we will discuss some basic facts on computable metric spaces and computable Banach spaces in the follow- ing section. Section 3 will be devoted to a short introduction into hyperspace representations and, finally, Section 4 presents our results on representations of the set of linear bounded operators. 2. Computable Metric and Banach Spaces In this section we will briefly discuss computable metric spaces and com- putable Banach spaces. The notion of a computable Banach space will be the central notion for all following results. Computable metric spaces have been used in the literature at least since Lacombe [15]. Restricted to computable points they have also been studied by various authors [8, 12, 17, 22]. We con- sider computable metric spaces as special separable metric spaces, but on all points and not only restricted to computable points [25]. Pour-El and Richards have introduced a closely related axiomatic characterization of sequential com- putability structures for Banach spaces [18] which has been extended to metric spaces by Mori, Tsujii, and Yasugi [27]. Before we start with the definition of computable metric spaces we just mention that we will denote the open balls of a metric space (X,d) by B(x,ε) := {y ∈ X : d(x,y) < ε} for all x ∈ X, ε > 0 and correspondingly the closed balls by B(x,ε) := {y ∈ X : d(x,y) ≤ ε}. Occasionally, we denote complements of sets A ⊆ X by Ac := X \A. Definition 2.1 (Computable metric space). A tuple (X,d,α) is called a com- putable metric space, if (1) d : X ×X → R is a metric on X, (2) α : N → X is a sequence which is dense in X, (3) d◦ (α×α) : N2 → R is a computable (double) sequence in R. Here, we tacitly assume that the reader is familiar with the notion of a computable sequence of reals, but we will come back to that point below. Oc- casionally, we will say for short that X is a computable metric space. Obviously, a computable metric space is especially separable. Given a computable metric space (X,d,α), its Cauchy representation δX :⊆ Σω → X can be defined by δX(01 n0+101n1+101n2+1 . . .) := lim i→∞ α(ni) for all ni such that (α(ni))i∈N converges and d(α(ni),α(nj)) ≤ 2−i for all j > i (and undefined for all other input sequences). In the following we tacitly assume that computable metric spaces are represented by their Cauchy representations. If X is a computable metric space, then it is easy to see that d : X ×X → R becomes computable (see Proposition 3.2 in [3]). All Cauchy representations are admissible with respect to the corresponding metric topology. Effective representations 119 An important computable metric space is (R,dR,αR) with the Euclidean metric dR(x,y) := |x−y| and some numbering of the rational numbers Q, as αR〈i,j,k〉 := (i − j)/(k + 1). Here, 〈i,j〉 := 1/2(i + j)(i + j + 1) + j denotes Cantor pairs and this definition is extended inductively to finite tuples. For short we will occasionally write k := αR(k). In the following we assume that R is endowed with the Cauchy representation δR induced by the computable metric space given above. This representation of R can also be defined, if (R,dR,αR) just fulfills (1) and (2) of the definition above and this leads to a definition of computable real number sequences without circularity. Occasionally, we will also use the represented space (Q,δQ) of rational numbers with δQ(1n0ω) := αR(n) = n. Many important representations can be deduced from computable metric spaces, but we will also need some differently defined representations. For instance, we will use two further representations ρ< , ρ> of the real numbers, which correspond to weaker information on the represented real numbers. Here ρ<(01 n0+101n1+101n2+1 . . .) = x : ⇐⇒ {q ∈ Q : q < x} = {ni : i ∈ N} and ρ< is undefined for all other sequences. Thus, ρ<(p) = x, if p is a list of all rational numbers smaller than x. Analogously, ρ> is defined with “>” in- stead of “<”. We write R< = (R,ρ<) and R> = (R,ρ>) for the corresponding represented spaces. The computable numbers in R< are called left-computable real numbers and the computable numbers in R> right-computable real num- bers. The representations ρ< and ρ> are admissible with respect to the lower and upper topology on R, which are induced by the open intervals (q,∞) and (−∞,q), respectively. Computationally, we do not have to distinguish the complex numbers C from R2. Thus, we can directly define a representation of C by δC := δ2R. If z = a + ib ∈ C, then we denote by z := a − ib ∈ C the conjugate complex number and by |z| := √ a2 + b2 the absolute value of z. Alternatively to this ad hoc definition of δC, we could consider δC as the Cauchy representation of a computable metric space (C,dC,αC), where αC is a numbering of Q[i], defined by αC〈n,k〉 := n + ki and dC(w,z) := |w − z| is the Euclidean metric on C. The corresponding Cauchy representation is equivalent to δ2 R . In the following we will consider vector spaces over R, as well as over C. We will use the notation F for a field which always might be replaced by both, R or C. Correspondingly, we use the notation (F,dF,αF) for a computable metric space which can be replaced by either of the computable metric spaces (R,dR,αR), (C,dC,αC) defined above. We will also use the notation QF = range(αF), i.e. QR = Q and QC = Q[i]. For the definition of a computable Banach space it is helpful to have the notion of a computable vector space, which we will define next. Definition 2.2 (Computable vector space). A represented space (X,δ) is called a computable vector space (over F), if (X, + , · , 0) is a vector space over F such that the following conditions hold: 120 Vasco Brattka (1) + : X ×X → X, (x,y) 7→ x + y is computable, (2) · : F×X → X, (a,x) 7→ a ·x is computable, (3) 0 ∈ X is a computable point. Here, (F,δF) is a computable vector space and if (X,δ) is a computable vec- tor space over F, then (Xn,δn) and (XN,δN) are computable vector spaces over F. If, additionally, (X,δ), (Y,δ′) are admissibly represented second countable T0–spaces, then (C(Y,X), [δ′ → δ]) is a computable vector space over F. Here we tacitly assume that the vector space operations on product, sequence and function spaces are defined componentwise. The proof for the function space is a straightforward application of evaluation and type conversion. The cen- tral definition for the present investigation will be the notion of a computable normed space. Definition 2.3 (Computable normed space). A tuple (X,‖ ‖,e) is called a computable normed space, if (1) ‖ ‖ : X → R is a norm on X, (2) e : N → X is a fundamental sequence, i.e. its linear span is dense in X, (3) (X,d,αe), with d(x,y) := ‖ x−y ‖ and αe〈k,〈n0, . . . ,nk〉〉 := ∑k i=0 αF(ni)ei, is a computable metric space with Cauchy representation δX, (4) (X,δX) is a computable vector space over F. If, in the situation of the definition, the underlying space (X,‖ ‖) is actually a Banach space, i.e. if (X,d) is a complete metric space, then (X,‖ ‖,e) is called a computable Banach space. If the norm and the fundamental sequence are clear from the context, or locally irrelevant, we will say for short that X is a computable normed space or a computable Banach space. We will always assume that computable normed spaces are represented by their Cauchy representations, which are admissible with respect to the norm topology. If X is a computable normed space, then ‖ ‖: X → R is a com- putable function. Of course, all computable Banach spaces are separable. In the following proposition a number of computable Banach spaces are defined. Proposition 2.4 (Computable Banach spaces). Let p ∈ R be a computable real number with 1 ≤ p < ∞ and let a < b be computable real numbers. The following spaces are computable Banach spaces over F. (1) (Fn,‖ ‖p,e) and (Fn,‖ ‖∞,e) with • ‖ (x1,x2, . . . ,xn) ‖p := p √ n∑ k=1 |xk|p and ‖ (x1,x2, . . . ,xn) ‖∞ := max k=1,...,n |xk|, • ei = e(i) = (ei1,ei2, . . . ,ein) with eik := { 1 if (∃j) i = jn + k, 0 otherwise. (2) (`p,‖ ‖p,e) with • `p := {x ∈ FN : ‖ x ‖p< ∞}, Effective representations 121 • ‖ (xk)k∈N ‖p := p √ ∞∑ k=0 |xk|p, • ei = e(i) = (eik)k∈N with eik := { 1 if i = k, 0 otherwise. (3) (C[a,b],‖ ‖,e) with • C[a,b] := {f : [a,b] → F | f continuous}, • ‖ f ‖ := max t∈[a,b] |f(t)|, • ei(t) = e(i)(t) = ti. We leave it to the reader to check that these spaces are actually computable Banach spaces. Unless stated otherwise, we will assume that (Fn,‖ ‖) is en- dowed with the maximum norm ‖ ‖ = ‖ ‖∞. It is known that the Cauchy representation δC[a,b] of C[a,b] = C([a,b],R) is equivalent to [δ[a,b] → δR], where δ[a,b] denotes the restriction of δR to [a,b] (cf. Lemma 6.1.10 in [26]). In the following we will occasionally utilize the sequence spaces `p to construct coun- terexamples. We close this section with a brief discussion of product spaces of computable normed spaces. Proposition 2.5 (Product spaces). If (X,‖ ‖,e), (Y,‖ ‖′,e′) are computable normed spaces, then the product space (X ×Y,‖ ‖′′,e′′), defined by ‖ (x,y) ‖′′ := max{‖ x ‖,‖ y ‖′} and e′′〈i,j〉 := (e(i),e′(j)), is a computable normed space too and the canonical projections of the product space pr1 : X ×Y → X and pr2 : X ×Y → Y are computable. 3. Hyperspaces of closed subsets Since we want to use the hyperspace A(X) of closed subsets A ⊆ X in order to represent linear bounded operators, we have to discuss representations of hyperspaces. Such representations have been studied in the Euclidean case in [6, 26] and for the metric case in [5]. Definition 3.1 (Hyperspace of closed subsets). Let (X,d,α) be a computable metric space. We endow the hyperspace A(X) := {A ⊆ X : A closed} of closed subsets with the representation δA(X), defined by δ>A(X)(01 〈n0,k0〉+101〈n1,k1〉+101〈n2,k2〉+1 . . .) := X \ ∞⋃ i=0 B(α(ni),ki). Whenever we have two representations δ,δ′ of some set, we can define the infimum δ u δ′ of δ and δ′ by (δ u δ′)〈p,q〉 = x : ⇐⇒ δ(p) = x and δ′(q) = x. We use the short notations A< = A<(X) = (A<(X),δ = A>(X) = (A>(X),δ>A(X)) and A = A(X) = (A(X),δ < A(X) uδ > A(X)) for the corresponding 122 Vasco Brattka represented spaces. For the computable points of these spaces special names are used. Definition 3.2 (Recursively enumerable and recursive sets). Let X be a com- putable metric space and let A ⊆ X be a closed subset. (1) A is called r.e. closed, if A is a computable point in A<(X), (2) A is called co-r.e. closed, if A is a computable point in A>(X), (3) A is called recursive closed, if A is a computable point in A(X). These definitions generalize the classical notions of r.e. and recursive sets, since a set A ⊆ N, considered as a closed subset of R, is r.e., co-r.e., recursive closed, if and only if it has the same property in the classical sense as a subset of N [6]. We close this section with a helpful result on hyperspaces, which follows directly from results in [5] and which can be considered as an effective version of the statement that closed subsets of separable metric spaces are separable. Here and in the following, A denotes the topological closure of a subset A ⊆ X of some topological space X. Proposition 3.3 (Separable closed subsets). Let X be a computable metric space. The mapping range : XN → A<(X), (xn)n∈N 7→ {xn : n ∈ N} is com- putable and if X is complete, then it admits a computable multi-valued partial right-inverse ⊆A<(X) ⇒ XN, defined for all non-empty closed subsets. 4. Representations of the Operator Space In this section we will define and compare several representations of the set B(X,Y ) of linear bounded operators T : X → Y on computable normed spaces X and Y . Definition 4.1 (Representations of the operator space). Let (X,‖ ‖,e) and Y be computable normed spaces. We define representations of B(X,Y ): (1) δev(p) = T : ⇐⇒ [δX → δY ](p) = T , (2) δgraph(p) = T : ⇐⇒ δA(X×Y )(p) = graph(T), (3) δ of B(X,Y ) by (1) δ=〈p,q〉 = T : ⇐⇒ δ(p) = T and δR(q) = ‖ T ‖, (2) δ>〈p,q〉 = T : ⇐⇒ δ(p) = T and δR(q) ≥‖ T ‖. All the representations introduced are admissible with respect to certain topolo- gies on the space of linear bounded operators (see the brief discussion in the Conclusion). While these representations separate into several distinct topo- logical and computational equivalence classes, the corresponding computable Effective representations 123 linear bounded operators do essentially coincide (see Corollary 4.6). This fol- lows from the following characterization which we have proved in [3]. Theorem 4.2 (Computable Linear Operators). Let X,Y be computable Ba- nach spaces, let (ei)i∈N be a computable sequence in X whose linear span is dense in X and let T : X → Y be a linear operator. Then the following conditions are equivalent: (1) T : X → Y is computable, (2) T : X → Y is bounded and maps computable sequences in X to com- putable sequences in Y, (3) T : X → Y is bounded and (Tek)k∈N is a computable sequence in Y , (4) graph(T) is an r.e. closed subset of X ×Y , (5) graph(T) is a recursive closed subset of X ×Y . The equivalence in this theorem is “non-constructive” in the sense that noth- ing is said about the possibility of converting one type of information into an- other type effectively. One of the main purposes of this paper is to study these possibilities. In this sense the following result includes a uniform version of the previous theorem. Theorem 4.3 (Representations of the operator space). Let X and Y be com- putable Banach spaces. Then the following reductions for representations of B(X,Y ) hold: (1) δ=ev ≤ δev ≤ δseq ≤ δ < graph and δev ≤ δgraph ≤ δ < graph, (2) δev ≡ δ>ev ≡ δ>seq ≡ δ <> graph ≡ δ > graph, (3) δ=ev ≡ δ=seq ≡ δ <= graph ≡ δ = graph. Proof. (1) “δ=ev ≤ δev” holds obviously and “δev ≤ δseq” follows by the evalua- tion property. “δseq ≤ δev” follows from Theorem 9.10 in [3] (see also Theorem 4.1 in [4]) which states that for every T ∈ B(X,Y ), given with respect to δev, we can effectively find some upper bound s ≥‖ T ‖. “δ>ev ≤ δev” obviously holds. “δ>ev ≤ δ>seq ≤ δ <> graph” follows directly from (1). “δ<>graph ≤ δ > ev” Given the graph(T) ∈ A<(X × Y ) of some linear bounded operator T : X → Y , we can effectively find a sequence f : N → X × Y such that range(f) is dense in graph(T) by Proposition 3.3. By Proposition 2.5 we can effectively find the projections f1 : N → X and f2 : N → Y of f too. Given x ∈ X, a real number s ≥ ‖ T ‖ and a precision m ∈ N we can effectively find n,k ∈ N and numbers q0, . . . ,qn ∈ QF such that s ≤ 2k and ‖ ∑n i=0 qif1(i) −x ‖ < 2 −m−k−1 since range(f1) is dense in X. It follows that∣∣∣∣∣ ∣∣∣∣∣T ( n∑ i=0 qif1(i) ) −T(x) ∣∣∣∣∣ ∣∣∣∣∣ ≤ ‖ T ‖ · ∣∣∣∣∣ ∣∣∣∣∣ n∑ i=0 qif1(i) −x ∣∣∣∣∣ ∣∣∣∣∣ < 2−m−1. By linearity of T we obtain T( ∑n i=0 qif1(i)) = ∑n i=0 qiTf1(i) = ∑n i=0 qif2(i) and thus we can evaluate T effectively up to any given precision m. Applying this idea, we can effectively find a Cauchy sequence which rapidly converges to T(x). Using type conversion we obtain the desired reducibility. “δ>ev ≤ δ > graph” and “δ > graph ≤ δ <> graph” follow from (1). (3) This follows directly from (2). � It should be mentioned that we have used completeness only for the reduc- tion “δ<>graph ≤ δ > ev” (implicitly, by application of Proposition 3.3) and thus for the results in (2) and (3), while the statement of (1) remains true for computable normed spaces X,Y . In case of a finite-dimensional Y = Fm, one can prove that the inverse graph−1 of the graph map graph : C(X,Fm) → A(X × Fm) is (δA(X×Fm), [δX → δFm])–computable, see Theorem 14.6 in [3]. As a direct consequence we obtain the following corollary. Corollary 4.4. Let X be a computable normed space and m ≥ 1 a natural number. Then δev ≡ δgraph for the corresponding representations of B(X,Fm). Now the question arises as to which of the reductions given in Theorem 4.3(1) are strict reductions. At least for certain spaces all of these reductions are strict as the following theorem shows. Figure 1 summarizes the results. δ=ev - δev δgraph δseq δ 1. “δev 6≤t δ=ev” In order to prove this, it suffices to show that the operator norm mapping ‖ ‖: B(`p,F) → R, defined for linear bounded T : `p → F, is not (δev,δR)–continuous. Let q be such that 1p + 1 q = 1 and q = ∞ if p = 1. For any sequence a = (ak)k∈N ∈ `q we define the functional λa : `p → F, (xk)k∈N 7→ ∑∞ k=0 akxk. In the following we will use the map L :⊆ F N×R → B(`p,F), (a,s) 7→ λa with dom(L) := {(a,s) ∈ `q ×R : ‖ a ‖q≤ s} and with q such that 1 p + 1 q = 1 and q = ∞ if p = 1. For all (a,s) ∈ dom(L) we obtain ‖ L(a,s) ‖= ‖ λa ‖= ‖ a ‖q≤ s and thus L is ([δNF ,δR],δev)–continuous. If we assume that the operator norm mapping ‖ ‖ is (δev,δR)–continuous, then this implies that ‖ ‖ ◦L :⊆ FN ×R → R is ([δN F ,δR],δR)–continuous. In particular, this implies that N :⊆ FN → R,a 7→‖ a ‖q is continuous for all a with ‖ a ‖q≤ 1. But this is obviously not the case (in general, no finite prefix of the sequence a determines the value ‖ a ‖q up to a given precision). “δseq 6≤t δev” Let us assume that δseq ≤t δev. Let L, λ and q be as defined above. Since λaei = ai for all a = (an)n∈N ∈ `q, it follows by the evaluation property and Theorem 9.10 in [3] (cf. Theorem 4.1 in [4]) that L admits a (δev, [δNF ,δR])–continuous multi-valued right-inverse L −1 :⊆B(`p,F) ⇒ FN×R. We consider the function λ :⊆ RN →B(`p,F), a 7→ λa with dom(λ) := RN∩`q. Since λaei = ai for all a = (an)n∈N ∈ `q, it also follows that λ is (δNR,δseq)– computable and thus (δN R ,δev)–continuous by assumption. Thus, L−1 ◦ λ :⊆ R N ⇒ FN ×R is (δN R , [δN F ,δR])–continuous too and hence the projection S :⊆ R N ⇒ R on the second component is continuous in the ordinary sense too. Thus S is a continuous operation such that there exists some s ∈ S(a) for all a ∈ RN with ‖ a ‖q< ∞ and for all such s the inequality ‖ a ‖q≤ s holds. Such a continuous operation can obviously not exist (since it would have to determine an upper bound of ‖ a ‖q, knowing only a finite prefix of the sequence a). A contradiction! “δseq 6≤t δgraph” This follows from δseq 6≤t δev since δgraph ≡ δev in the case of B(`p,F) by Corollary 4.4. “δev) does not lead to a strictly stronger representation while Theorem 4.5 shows that adding the operator bound itself actually leads to a stronger type of information. Representing linear bounded operators by a sequence of values on a fundamental sequence (i.e. δseq) and representing operators by positive and negative information on their graphs (i.e. δgraph) both lead to mutually incomparable but weaker representations than δev. But both types of information are strictly stronger than just positive information on the operator graph (i.e. δ