Filter-dependent versions of the Uniform Boundedness Principle Ben De Bondt∗† Hans Vernaeve‡ Abstract For every filter F on N; we introduce and study corresponding uniform F-boundedness principles for locally convex topological vector spaces. These principles generalise the classical uniform boundedness principles for se- quences of continuous linear maps by coinciding with these principles when the filter F equals the Fr´echet filter of cofinite subsets of N: We determine combinatorial properties for the filter F which ensure that these uniform F-boundedness principles hold for every Fr´echet space. Furthermore, for several types of Fr´echet spaces, we also isolate properties of F that are necessary for the validity of these uniform F-boundedness principles. For every infinite-dimensional Banach space X, we obtain in this way exact combinatorial characterisations of those filters F for which the correspond- ing uniform F-boundedness principles hold true for X. Keywords: Uniform Boundedness Principle, Filters on N, Fr´echet spaces, Banach spaces MSC2010: 46A04, 03E20 (Primary), 40A35, 46B99 (Secondary) arXiv:2001.01663v2 [math.FA] 2 Nov 2020 ∗Ben De Bondt gratefully acknowledges support by Ghent University, through BOF PhD grant BOF18/DOC/298, which supported both the research and the preparation of the current article. †Currently at Universit´ede Paris, Institut de Math´ematiquesde Jussieu-Paris Rive Gauche, as a Cofund MathInParis PhD fellow. The Cofund MathInParis PhD project has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sk lodowska-Curie grant agreement No. 754362. ‡Hans Vernaeve gratefully acknowledges support by research Grant 1.5.129.18N of the Research Foundation Flanders FWO. 1 1 Introduction This paper contributes to the line of research ([2], [3], [7], [10], [12], [17], [18], . ) that is concerned with the study of filter versions1 of several the- orems in analysis (or infinite combinatorics), whose classical proofs critically rely on Baire-categoric, measure-theoretic, or Ramsey-theoretic ideas. Before clarifying how to derive in a canonical way the statement TF for a filter version of the theorem T , let us quickly motivate the interest in the new statement TF (see also the introduction of [18]). As well as in generalising the theorem T involved, additional motivation for the study of the filter version TF lies in gaining understanding of the combinatorial principles that are needed to prove T . Indeed, those filters on N that preserve validity of the filter versions often turn out to possess interesting combinatorial properties, closely related to the P-properties and Q-properties, that are classically studied in set theory. In the present paper, we show that this is true as well for filter versions of the uniform boundedness principle for Fr´echet spaces. One main corollary of the uniform boundedness principle entails continuity of the pointwise limits of sequences of continuous linear mappings. In [19] it was shown that one particular filter version of this consequence of the UBP holds for separable Banach spaces whenever the dual ideal has the property denoted in [19] by property (APO). Another important consequence of the UBP (weak convergence implies bound- edness in Banach spaces) was studied from the filter perspective in [7], [12] (see Remark 3.3 below). In [3], Avil´es,Kadets, P´erezand Solecki studied the Baire filters2, which have the defining property that for every complete metric space X and every order- reversing map f :(F; ⊆) ! (fF ⊆ X : F nowhere dense in Xg; ⊆); one has that [ f(A) 6= X: A2F Given that the uniform boundedness principle comes as a consequence of the Baire category theorem, it is not surprising that the uniform F-boundedness principle will hold for every Fr´echet space whenever F is a (free) Baire filter (see Lemma 1.1 below). However, it was found in [3], that the notion of Baire filter is a rather restrictive one. The analytic Baire filters for instance, are all countably generated. This makes it natural to determine weaker conditions under which filter versions of the uniform boundedness principle are true.3 1Because of the duality between filters and ideals, one can equivalently study the corres- ponding ideal versions of such theorems. Several authors choose to do so. 2Or rather, the ideals dual to such filters. 3In particular, Example 5.1 will exhibit an analytic filter of uncountable character for which the uniform F-boundedness principle holds for every Fr´echet space. 2 1.1 Terminology and notation We now proceed to the introduction of the key concepts under consideration. Every filter F on N gives rise to the generalised quantifier (8F i 2 N) defined by the following rule: (8F i 2 N)(Φ(i)) , fi 2 N : Φ(i)g 2 F; for every formula Φ: We let C be the Fr´echet filter consisting of the cofinite subsets of N and let F be an arbitrary filter on N: Then, any concept which can be expressed in terms of the quantifier (8C i 2 N) has a direct F-analogue, obtained by replacing the quantifier (8C i 2 N) by the quantifier (8F i 2 N): For example, convergence of a sequence (xi)i in a metric space (M; d) to x 2 M can be expressed by the formula (8" 2 R>0)(8C i 2 N) d(x; xi) < "; and the replacement of generalised quantifiers gives the useful concept of F-con- vergence of the sequence (xi)i to x: (8" 2 R>0)(8F i 2 N) d(x; xi) < ": In the same fashion, we now define F-analogues for two principal concepts from the theory of topological vector spaces, that of boundedness of a sequence and that of equicontinuity of a sequence of (continuous linear) maps. In what follows, all sequences are indexed by the non-negative integers, unless stated otherwise. Let X be a topological vector space and F a filter on N. A sequence (xi)i in X is F-bounded if for every 0-neighbourhood U in X there exists t > 0 such that txi 2 U (8F i 2 N): Note that in a normed space X,(xi)i is F-bounded if and only if there exists C > 0 such that kxik ≤ C (8F i 2 N): Also note that for general filters F, F-boundedness is not necessarily preserved by rearrangements of sequences. Given two topological vector spaces X; Y , let L(X; Y ) denote the vector space of continuous linear maps from X to Y . A sequence (Ti)i in L(X; Y ) is F-equicontinuous if for every 0-neighbourhood V in Y there exists a 0-neighbourhood U in X such that Ti[U] ⊆ V (8F i 2 N): The reader can observe that if X and Y are both normed spaces, a sequence (Ti)i of continuous linear maps from X to Y is F-equicontinuous precisely when it is F-bounded in the normed space L(X; Y ) of continuous linear operators from X to Y: Continuing in this same vein, a sequence (Ti)i in L(X; Y ) is defined to be point- wise F-bounded when for every x 2 X; the sequence (Ti(x))i is an F-bounded 3 sequence in Y: Equivalently, (Ti)i is pointwise F-bounded when it is F-bounded in the space L(X; Y ) equipped with the topology of pointwise convergence. Given the topological vector spaces X; Y and the filter F on N; let PWF (X; Y ) and EQF (X; Y ) denote respectively the set of all pointwise F-bounded se- quences in L(X; Y ) and the set of all F-equicontinuous sequences in L(X; Y ). It is clear that EQF (X; Y ) ⊆ PWF (X; Y ): The validity of the reverse inclusion is the subject of uniform F-boundedness principles. We write UBPF (X; Y ) as abbreviation for the statement PWF (X; Y ) = EQF (X; Y ) and say that the uniform F-boundedness principle holds for continuous linear maps from X to Y whenever this statement is true. We will also say that the uniform F-bound- edness principle holds for X if UBPF (X; Y ) holds for every locally convex space Y: We define F to be a Fr´echet-UBP-filter if UBPF (X; Y ) holds for every Fr´echet space X and every locally convex space Y: By the classical theo- rem of Banach and Steinhaus, the Fr´echet filter C is a Fr´echet-UBP-filter. We define F to be a Banach-UBP-filter if UBPF (X; Y ) holds for every Ba- nach space X and every locally convex space Y: The key concepts being introduced above, we can now state our main objectives. Before doing so, we briefly illustrate the concepts just introduced by checking that the Baire filters studied in [3] are indeed Fr´echet-UBP-filters. Lemma 1.1. If the filter F (on N) is a Baire filter containing every cofinite subset of N, then UBPF (X; Y ) holds for every Fr´echetspace X and every topological vector space Y: Proof. Let (Ti)i be a pointwise F-bounded sequence in L(X; Y ). We mimic the Baire category proof of the uniform boundedness principle to prove that (Ti)i is an F-equicontinuous sequence. Let W be an arbitrary 0-neighbourhood in Y and choose a second 0-neighbourhood V in Y which is closed and balanced, and satisfies V − V ⊆ W . For every F 2 F and n 2 N, we define a corresponding set CF;n ⊆ X as follows: \ −1 CF;n := Ti [nV ]: i2F Then, the pointwise F-boundedness of (Ti)i allows us to write: [ [ [ X = CF;n = CF;min(F ): F 2F n2N F 2F Since f :(F; ⊆) ! (P (X); ⊆): F 7! CF;min(F ) is an order-reversing map with CF;min(F ) closed for any F 2 F, it follows from the defining property of Baire filters that for some F 2 F, the set CF;min(F ) has non-empty interior.