## General logical metatheorems for functional analysis (2008)

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Citations: | 30 - 19 self |

### BibTeX

@MISC{Gerhardy08generallogical,

author = {Philipp Gerhardy and Ulrich Kohlenbach},

title = { General logical metatheorems for functional analysis},

year = {2008}

}

### OpenURL

### Abstract

In this paper we prove general logical metatheorems which state that for large classes of theorems and proofs in (nonlinear) functional analysis it is possible to extract from the proofs effective bounds which depend only on very sparse local bounds on certain parameters. This means that the bounds are uniform for all parameters meeting these weak local boundedness conditions. The results vastly generalize related theorems due to the second author where the global boundedness of the underlying metric space (resp. a convex subset of a normed space) was assumed. Our results treat general classes of spaces such as metric, hyperbolic, CAT(0), normed, uniformly convex and inner product spaces and classes of functions such as nonexpansive, Hölder-Lipschitz, uniformly continuous, bounded and weakly quasinonexpansive ones. We give several applications in the area of metric fixed point theory. In particular, we show that the uniformities observed in a number of recently found effective bounds (by proof theoretic analysis) can be seen as instances of our general logical results.

### Citations

398 |
Metric Spaces of Non-Positive Curvature
- Bridson, Haefliger
- 1999
(Show Context)
Citation Context ...yperbolic type’ although it differs from the definition of the latter in [11].sGENERAL LOGICAL METATHEOREMS FOR FUNCTIONAL ANALYSIS 2619 CAT(0)-spaces having the geodesic line extension property (see =-=[3]-=- and below for details on CAT(0)-spaces). As carried out in detail in [25] we formalize our classes of spaces on top of a formal system A ω of classical analysis which is based on a language of functi... |

154 |
Über eine bisher noch nicht benützte Erweiterung des finiten Standpunkts
- Gödel
- 1958
(Show Context)
Citation Context ...s 0 and X, i.e. (i) 0,X ∈ T X , (ii) ρ, τ ∈ T X ⇒ (ρ → τ) ∈ T X (in particular, the constants Πρ,τ , Σδ,ρ,τ ,R ρ for λ-abstraction and simultaneous primitive recursion (in the extended sense of Gödel =-=[10]-=-) and their defining axioms and the schemes IA (induction), QF-AC (quantifier-freesGENERAL LOGICAL METATHEOREMS FOR FUNCTIONAL ANALYSIS 2621 choice in all types), DC (dependent countable choice) 4 and... |

102 |
Mean value methods in iterations
- Mann
- 1953
(Show Context)
Citation Context ...n arbitrary (nonempty) hyperbolic space, k ∈ N, k≥ 1, and (λn)n∈N a sequence in [0, 1 − 1 ∞� k ]with λn = ∞, and define for n=0 f : X → X, x ∈ X the Krasnoselski-Mann iteration (xn)n starting from x (=-=[30, 35]-=-) by x0 := x, xn+1 := (1 − λn)xn ⊕ λnf(xn). In [11] (Theorem 1) the following is proved21 ∀x ∈ X, f : X → X � (xn)n bounded and f n.e. → lim n→∞ d(xn,f(xn)) = 0 � . As observed in [2], it actually suf... |

73 |
Provably recursive functionals of analysis: a consistency proof of analysis by an extension of principles formulated in current intuitionistic mathematics
- Spector
- 1962
(Show Context)
Citation Context ...hat A ω [X, d]−b ⊢∀x ρ (∀u 0 B∀(x, u) →∃v 0 C∃(x, v)). Then there exists a partial functional Φ:S�ρ ⇀ N whose restriction to the strongly majorizable elements M�ρ of S�ρ 9 is a total (bar recursively =-=[40]-=-) 8 Here we use the plural since the interpretation of 0X is not uniquely determined. 9 In the sense of [1].sGENERAL LOGICAL METATHEOREMS FOR FUNCTIONAL ANALYSIS 2627 computable functional 10 and the ... |

58 |
Recursive functionals and quantifiers of finite types
- Kleene
- 1963
(Show Context)
Citation Context ...are their own functional interpretation. Again the extracted bounds depend on (a-majorants for) these new constants. Then the conclusion holds in all metric (X, d), resp. hyperbolic 10In the sense of =-=[16]-=- relativized to the type structure Mω of strongly majorizable functionals from [1]. 11Note that x∗ �a x implies that x∗ s-maj �ρ x∗ and hence the strong majorizability of x∗ so that Φ(x∗ ) is defined.... |

56 | Analysing proofs in analysis, in
- Kohlenbach
- 1993
(Show Context)
Citation Context ...mal form of a slightly different but trivially equivalent formulation of the Cauchy property. 47Corollary 4.14 as well as the proof-theoretic study of the Bolzano-Weierstraß principle carried out in =-=[17]-=-. This concrete Ω even is independent from b and is defined as follows Ω(l, g, δ, γ) := maxΨ0(i, l, g, δ), i≤γ(l+3) where ⎧⎨ ⎩ Ψ0(0,l,g,δ):=0 Ψ0(n+1,l,g,δ):=δ ( l+2+⌈log2(max i≤n g(Ψ0(i, ) l, g, δ)) +... |

47 |
Groupes réductifs sur un corps local, I. Données radicielles valuées
- Bruhat, Tits
- 1972
(Show Context)
Citation Context ...(x, z, λ),WX(y, w, λ)) ≤R (1R −R ˜ λ)dX(x, y)+R ˜ λdX(z, w) � . Aω [X, d, W,CAT(0)]−b results from Aω [X, d, W]−b by adding as a further axiom the formalized form of the Bruhat-Tits or CN−-inequality =-=[4]-=-, i.e. ∀x X ,y X 1,y X� 2 dX(x,WX(y1,y2, 1 2 ))2 1 ≤R 2 dX(x, y1) 2 1 +R 2 dX(x, y2) 2 1 −R 4 dX(y1,y2) 2� . Remark 2.6. (1) The additional axioms of Aω [X, d]−b express (modulo our representation of ... |

44 | Hereditarily majorizable functionals of finite type, Metamathematical investigation of intuitionistic Arithmetic and Analysis - Howard - 1973 |

38 |
Fixed points and iteration of a nonexpansive mapping in a Banach space
- Ishikawa
- 1976
(Show Context)
Citation Context ...a computable bound Φ(k, α, b, l) such that in any (nonempty) hyperbolic space (X, d, W), for any 21 For the case of convex subsets C ⊆ X of normed linear spaces (X, �·�) this result is already due to =-=[13]-=-. [11] even treats spaces of hyperbolic type.s2646 PHILIPP GERHARDY AND ULRICH KOHLENBACH l, b, k ∈ N and any α : N → N the following holds: if (λn) is a sequence in [0, 1) such that ∀n ∈ N(λn ≤ 1 − 1... |

36 |
Strongly majorizable functionals of finite type: a model of bar recursion containing discontinuous functionals
- Bezem
- 1985
(Show Context)
Citation Context ...(including CAT(0)-spaces), as well as normed spaces. By a hyperbolic space we understand the following: Definition 2.1. (X, d, W) is called a hyperbolic space if (X, d) isametricspace and W : X × X × =-=[0, 1]-=- → X a function satisfying (i) ∀x, y, z ∈ X∀λ ∈ [0, 1] � d(z, W(x, y, λ)) ≤ (1 − λ)d(z, x)+λd(z, y) � , (ii) ∀x, y ∈ X∀λ1,λ2 ∈ [0, 1] � d(W(x, y, λ1),W(x, y, λ2)) = |λ1 − λ2|·d(x, y) � , (iii) ∀x, y ∈... |

31 | Effective moduli from ineffective uniqueness proofs. An unwinding of de La Vallée Poussin’s proof for Chebycheff approximation
- Kohlenbach
- 1993
(Show Context)
Citation Context ...z, v free, where furthermore 0X does not occur in B∀,C∃.If A ω [X, d, W]−b ⊢∀x∈P∀y∈K∀z τ (∀u 0 B∀ →∃v 0 C∃), 10 See [22] for an extensive discussion of this point. 11 For details on this see [22] and =-=[16]-=-. 19then there exists a computable functional Φ:IN IN × IN (IN×...×IN) → IN s.t. the following holds in every nonempty hyperbolic space (X, d, W): for all representatives rx ∈ IN IN of x ∈ P and all ... |

28 |
I.: Krasnoselski-Mann iterations in normed spaces
- Borwein, Reich, et al.
- 1992
(Show Context)
Citation Context ...ing from x ([30, 35]) by x0 := x, xn+1 := (1 − λn)xn ⊕ λnf(xn). In [11] (Theorem 1) the following is proved21 ∀x ∈ X, f : X → X � (xn)n bounded and f n.e. → lim n→∞ d(xn,f(xn)) = 0 � . As observed in =-=[2]-=-, it actually suffices to assume that (x ∗ n)n starting from some x ∗ is bounded. Therefore ∀x∈X, f : X → X � ∃x∗ ∈ X((x∗ n)n bounded) and f n.e. → lim n→∞ d(xn,f(xn)) = 0 � . The proof given in [11] ... |

28 |
Extensional Gödel Functional Interpretation
- Luckhardt
(Show Context)
Citation Context ...y even without DC and in a quantifier-free fragment of this theory) augmented by the schema (BR) of simultaneous bar recursion in all types of T X (for A ω this fundamental result is due to [40] (see =-=[33]-=- for a comprehensive treatment) which extends [10], where functional interpretation was introduced).sGENERAL LOGICAL METATHEOREMS FOR FUNCTIONAL ANALYSIS 2653 Let A ω [X, d, W] − −b := Aω [X, d, W]−b ... |

24 |
Iteration processes for nonexpansive mappings
- Goebel, Kirk
(Show Context)
Citation Context ... discussed in detail in [22], we obtain Takahashi’s [34] ‘convex metric spaces’ if the axioms (ii)-(iv) are dropped, and a notion which is equivalent to the concept of ‘space of hyperbolic type’ from =-=[11]-=- if we drop only (iv). As observed in [31, 2] and [32], several arguments in metric fixed point theory require a bit more of linear structure which gave rise to a notion of ‘hyperbolic space’ 1 in [14... |

21 | A quantitative version of a theorem due to Borwein-Reich-Shafrir
- Kohlenbach
(Show Context)
Citation Context ... arbitrary types. The theorems were applied to results in metric fixed point theory to explain the extractability of strong uniform bounds that had been observed previously in several concrete cases (=-=[21, 23, 28]-=-) as well as to predict new such bounds 1which subsequently could, indeed, be found following the extraction algorithm provided by monotone functional interpretation ([26, 24]). In the concrete appli... |

16 |
On the Mann iterative process
- Dotson
- 1970
(Show Context)
Citation Context ... 1andforallx, y ∈ X. For normed linear spaces (X, �·�) those definitions are to be understood w.r.t. the induced metric d(x, y):=�x− y�. The notion of quasi-nonexpansivity was introduced by Dotson in =-=[6]-=-, and the notion of weak quasi-nonexpansivity is (implicitly) due to B. Lambov and the second author [26] (note that in contexts where quasi-nonexpansive mappings are used it is always assumed that fi... |

14 |
Nonexpansive mappings, asymptotic regularity and successive approximations
- Edelstein, O’Brien
- 1978
(Show Context)
Citation Context ...hown to follow from a general logical metatheorem in [25] where a detailed 22 For the case of bounded convex subsets of normed spaces and constant λn = λ ∈ (0, 1) the uniformity in x wasalreadyshownin=-=[7]-=-and–for(λn)n in [a, b] ⊂ (0,1) and non-increasing – in [5]. i=0sGENERAL LOGICAL METATHEOREMS FOR FUNCTIONAL ANALYSIS 2647 discussion of this point is given. In [28], the extraction of an actual effect... |

14 |
Krasnoselskii’s Iteration process in hyperbolic space
- Kirk
- 1981
(Show Context)
Citation Context ...11] if we drop only (iv). As observed in [31, 2] and [32], several arguments in metric fixed point theory require a bit more of linear structure which gave rise to a notion of ‘hyperbolic space’ 1 in =-=[14, 31]-=- which adds axiom (iv) (for λ := 1) and the 2 requirement that any two points not only are connected by a metric segment but by a metric line. As a consequence of this (just as in the case of normed s... |

13 | On the computational content of the Krasnoselski and Ishikawa point theorems
- Kohlenbach
- 2001
(Show Context)
Citation Context ...nding on the principles used in the given proof). The proof of the metatheorem provides an algorithm for actually extracting Ψ from a given proof and in many cases this has been carried out (see e.g. =-=[21, 22, 23, 24, 26, 31, 32]-=- and the discussion in section 8 below). 2 Definitions The classes of general spaces we are considering are metric spaces, hyperbolic spaces (including CAT(0)-spaces) as well as normed spaces. By a hy... |

12 |
Two remarks on the method of successive approximations
- Krasnoselski
- 1955
(Show Context)
Citation Context ...n arbitrary (nonempty) hyperbolic space, k ∈ N, k≥ 1, and (λn)n∈N a sequence in [0, 1 − 1 ∞� k ]with λn = ∞, and define for n=0 f : X → X, x ∈ X the Krasnoselski-Mann iteration (xn)n starting from x (=-=[30, 35]-=-) by x0 := x, xn+1 := (1 − λn)xn ⊕ λnf(xn). In [11] (Theorem 1) the following is proved21 ∀x ∈ X, f : X → X � (xn)n bounded and f n.e. → lim n→∞ d(xn,f(xn)) = 0 � . As observed in [2], it actually suf... |

10 | Strongly uniform bounds from semi-constructive proofs
- Gerhardy, Kohlenbach
(Show Context)
Citation Context ...even in the presence of many ineffective principles) be avoided and effective bounds for formulas C of arbitrary complexity can be extracted (though no longer bounds on universal premises ∀u0B∀). See =-=[9]-=- for this. As a corollary to the proof of Theorem 4.10 we obtain Theorem 3.7 in [25]: Corollary 4.15. (1) Let σ, ρ be types of degree 1 and τ be a type of degree (1,X). Let sσ→ρ be a closed term of Aω... |

10 | Proof mining: a systematic way of analyzing proofs in mathematics - Kohlenbach, Oliva - 2003 |

10 | Effective uniform bounds from proofs in abstract functional analysis
- Kohlenbach
- 2008
(Show Context)
Citation Context ...s well as their products are simultanously present. We are confident that these results will have many more applications also outside the context of fixed point theory (see [29] and – in particular – =-=[18]-=- for surveys of different topics to which this kind of ‘proof mining’ approach can be applied). Let us now sketch the general shape of the results we are going to prove in this paper: we work in an ap... |

9 |
I.: Nonexpansive iterations in hyperbolic spaces
- Reich, Shafrir
- 1990
(Show Context)
Citation Context ...kahashi’s [41] ‘convex metric spaces’ if the axioms (ii)-(iv) are dropped, and a notion which is equivalent to the concept of ‘space of hyperbolic type’ from [11] if we drop only (iv). As observed in =-=[37, 2]-=- and [38], several arguments in metric fixed point theory require a bit more linear structure which gave rise to a notion of ‘hyperbolic space’ 2 in [14, 37] which adds axiom (iv) (for λ := 1 2 ) and ... |

9 |
A quantitative version of a theorem due to
- Kohlenbach
(Show Context)
Citation Context ... arbitrary types. The theorems were applied to results in metric fixed point theory to explain the extractability of strong uniform bounds that had been observed previously in several concrete cases (=-=[18, 20, 24]-=-) as well as to predict new such bounds which subsequently could, indeed, be found following the extraction algorithm provided by monotone functional interpretation ([23, 21]). In the concrete applica... |

7 |
A logical uniform boundedness principle for abstract metric and hyperbolic spaces
- Kohlenbach
- 2006
(Show Context)
Citation Context ...junction of such premises. Remark 4.16. This result (first proved in [25]) was recently strengthened by extending the formal systems by a strong nonstandard uniform boundedness principle ∃-UB X ; see =-=[17]-=-. Proof. Take a =0X. For x, which has type σ of degree 1 (w.l.o.g. σ =1),we easily see (even using only strong majorization s-maj) that x M � 0X x.Next,for the 0X-majorant s ∗ � 0X s, which we can con... |

7 |
Nonexpansive mappings and asymptotic regularity, Nonlinear Analysis TMA 40
- Kirk
- 2000
(Show Context)
Citation Context ...t can now for the first time be explained by our refined logical metatheorems as well. Note that the proof of Theorem 2 in [11] (as well as the alternative proof for constant λn = λ ∈ (0, 1) given in =-=[15]-=-) crucially uses that the whole space X is assumed to be bounded. So the uniformity 20 For the case of bounded convex subsets of normed spaces and constant λn = λ ∈ (0, 1) the uniformity in x was alre... |

6 | Bounds on iterations of asymptotically quasinonexpansive mappings - Kohlenbach, Lambov - 2004 |

6 | L.: The approximate fixed point property in product spaces
- Kohlenbach, Leuştean
(Show Context)
Citation Context ...]). Our refined metatheorems for the first time allow one to explain this finding as an instance of a general result in logic. For functional analytic applications of the uniformity provided by Ψ see =-=[27]-=-. 25 For details see [23].s2650 PHILIPP GERHARDY AND ULRICH KOHLENBACH 9. Proofs of Theorems 4.10 and 6.3 We focus on proving Theorems 4.10 and 6.3 for the theories Aω [X, d, W]−b and Aω [X, �·�,C]−b,... |

6 |
A characterization of convex subsets of normed spaces
- Machado
- 1973
(Show Context)
Citation Context ...type X and not 0. Hence our corollary does not apply. 6. Metatheorems for normed linear spaces We now discuss the setting of (real) normed linear spaces with convex subsets C. As discussed in Machado =-=[34]-=-, one may characterize convex subsets of normed spaces in the setting of hyperbolic spaces in terms of additional conditions on the function W. The additional conditions are (I) that the convex combin... |

6 |
Nonexpansive mappings and asymptotic regularity
- Kirk
(Show Context)
Citation Context ...t can now for the first time be explained by our refined logical metatheorems as well. Note that the proof of Theorem 2 in [11] (as well as the alternative proof for constant λn = λ ∈ (0, 1) given in =-=[15]-=-) crucially uses that the whole space X is assumed to be bounded. So the uniformity result guaranteed a-priorily by the metatheorems of the present paper applied to Theorem 1 of [11] not only yields i... |

4 | On the approximation of fixed points of nonexpansive mappings
- Chidume
- 1981
(Show Context)
Citation Context ...where a detailed 22 For the case of bounded convex subsets of normed spaces and constant λn = λ ∈ (0, 1) the uniformity in x wasalreadyshownin[7]and–for(λn)n in [a, b] ⊂ (0,1) and non-increasing – in =-=[5]-=-. i=0sGENERAL LOGICAL METATHEOREMS FOR FUNCTIONAL ANALYSIS 2647 discussion of this point is given. In [28], the extraction of an actual effective uniform rate of convergence was carried out and it was... |

4 |
Iteration processes for nonexpansive mappings, Topological Methods in Nonlinear Functional Analysis
- Goebel, Kirk
- 1982
(Show Context)
Citation Context ... discussed in detail in [25], we obtain Takahashi’s [41] ‘convex metric spaces’ if the axioms (ii)-(iv) are dropped, and a notion which is equivalent to the concept of ‘space of hyperbolic type’ from =-=[11]-=- if we drop only (iv). As observed in [37, 2] and [38], several arguments in metric fixed point theory require a bit more linear structure which gave rise to a notion of ‘hyperbolic space’ 2 in [14, 3... |

3 |
S.Prus, The fixed point property for mappings admitting a center, Nonlinear Analysis
- Garcia-Falset
(Show Context)
Citation Context ...∃v 0 C∃(x, y, z, f, v) � , where 0X does not occur in B∀ and C∃. 17 The concept of weakly quasi-nonexpansive mapping has recently been formulated independently – under the name of J-type mapping – in =-=[8]-=- where the fixed point p is called a ‘center’.sGENERAL LOGICAL METATHEOREMS FOR FUNCTIONAL ANALYSIS 2631 Then there exists a computable functional Φ:NN × N → N s.t. for all representatives rx ∈ NN of ... |

3 |
The approximate fixed point property in Banach and hyperbolic spaces
- Shafrir
- 1990
(Show Context)
Citation Context ...1] ‘convex metric spaces’ if the axioms (ii)-(iv) are dropped, and a notion which is equivalent to the concept of ‘space of hyperbolic type’ from [11] if we drop only (iv). As observed in [37, 2] and =-=[38]-=-, several arguments in metric fixed point theory require a bit more linear structure which gave rise to a notion of ‘hyperbolic space’ 2 in [14, 37] which adds axiom (iv) (for λ := 1 2 ) and the requi... |

2 |
Examples of fixed point free mappings, Handbook of Metric Fixed Point Theory
- Sims
- 2001
(Show Context)
Citation Context ...paces always have ε-fixed points for arbitrary ε>0, while they need not have exact fixed points in general (not even in the case of bounded, closed and convex subsets of Banach spaces such as c0; see =-=[39]-=-). Hence, for bounded hyperbolic spaces and nonexpansive mappings the premise ‘∀ε >0Fixε(f,z,b) �= ∅’ can be dropped if b is taken as an upper bound on the metric d. For further discussion, see Remark... |

1 |
asymptotic regularity for Mann iterates
- Uniform
(Show Context)
Citation Context ... arbitrary types. The theorems were applied to results in metric fixed point theory to explain the extractability of strong uniform bounds that had been observed previously in several concrete cases (=-=[21, 23, 28]-=-) as well as to predict new Received by the editors March 17, 2006. 2000 Mathematics Subject Classification. Primary 03F10, 03F35, 47H09, 47H10. 2615 c○2007 American Mathematical Societys2616 PHILIPP ... |

1 |
iterates of directionally nonexpansive mappings in hyperbolic spaces,Abstract and Applied Analysis 8
- Mann
- 2003
(Show Context)
Citation Context ... arbitrary types. The theorems were applied to results in metric fixed point theory to explain the extractability of strong uniform bounds that had been observed previously in several concrete cases (=-=[21, 23, 28]-=-) as well as to predict new Received by the editors March 17, 2006. 2000 Mathematics Subject Classification. Primary 03F10, 03F35, 47H09, 47H10. 2615 c○2007 American Mathematical Societys2616 PHILIPP ... |

1 |
A quadratic rate of asympotic regularity for CAT(0)-spaces
- Leu¸stean
(Show Context)
Citation Context ...ding on the principles used in the given proof). The proof of the metatheorem provides an algorithm for actually extracting Ψ from a given proof, and in many cases this has been carried out (see e.g. =-=[21, 22, 23, 24, 26, 31, 32]-=- and the discussion in section 8 below). 2. Definitions The classes of general spaces we are considering are metric spaces, hyperbolic spaces (including CAT(0)-spaces), as well as normed spaces. By a ... |

1 |
mining in R-trees and hyperbolic spaces
- Proof
- 2006
(Show Context)
Citation Context ...ding on the principles used in the given proof). The proof of the metatheorem provides an algorithm for actually extracting Ψ from a given proof, and in many cases this has been carried out (see e.g. =-=[21, 22, 23, 24, 26, 31, 32]-=- and the discussion in section 8 below). 2. Definitions The classes of general spaces we are considering are metric spaces, hyperbolic spaces (including CAT(0)-spaces), as well as normed spaces. By a ... |

1 |
When uniformly-continuous implies bounded
- O’Farrell
(Show Context)
Citation Context ...ct intermediate points in order to be able to make use of the uniform continuity of f. For a study of metric spaces for which uniformly continuous functions f admit the definition of a suitable Ω see =-=[36]-=-. Otherwise, in the setting of metric spaces, we need to require explicitly that a given uniformly continuous function f with modulus ω also satisfies (∗∗) with a suitable Ω. As a generalization of Co... |

1 |
A convexity in metric space and nonexpansive mappings,I.KodaiMath.Sem. Rep. 22
- Takahashi
- 1970
(Show Context)
Citation Context ...λ ∈ [0, 1] � W(x, y, λ)=W(y, x,1 − λ) � � , ∀x, y, z, w ∈ X, λ ∈ [0, 1] (iv) � � d(W(x, z, λ),W(y, w, λ)) ≤ (1 − λ)d(x, y)+λd(z, w) . Remark 2.2. As discussed in detail in [25], we obtain Takahashi’s =-=[41]-=- ‘convex metric spaces’ if the axioms (ii)-(iv) are dropped, and a notion which is equivalent to the concept of ‘space of hyperbolic type’ from [11] if we drop only (iv). As observed in [37, 2] and [3... |