Orlicz–Pettis theorem
A theorem in functional analysis concerning convergent series (Orlicz) or, equivalently, countable additivity of measures (Pettis) with values in abstract spaces. Let formula_1 be a Hausdorff locally convex topological vector space with dual formula_2. A series formula_3 is "subseries convergent" (in formula_4), if all its subseries formula_5 are convergent. The theorem says that, equivalently, The history of the origins of the theorem is somewhat complicated. In numerous papers and books there are misquotations or/and misconceptions concerning the result. Assuming that formula_1 is weakly sequentially complete Banach space, W. Orlicz proved the following Theorem. If a series  formula_6 is weakly unconditionally Cauchy, i.e., formula_17 for each linear functional formula_18, then the series is (norm) convergent in formula_4. After the paper was published, Orlicz realized that in the proof of the theorem the weak sequential completeness of formula_1 was only used to guarantee the existence of the weak limits of the considered series. Consequently, assuming the existence of those limits, which amounts to the assumption of the weak subseries convergence of the series, the same proof shows that the series in norm convergent. In other words, the version (i) of the Orlicz–Pettis theorem holds. The theorem in  this form, openly credited to Orlicz, appeared in Banach's monograph in the last chapter "Remarques" in which no proofs were provided. Pettis directly referred to Orlicz's theorem in Banach's book. Needing the result in order to show the coincidence of the weak and strong measures, he provided a proof. Also Dunford gave a proof (with a remark that it is similar to the original proof of Orlicz). A more thorough discussion of the origins of the Orlicz–Pettis theorem and, in particular, of the paper can be found in. See also footnote 5 on p. 839 of  and the comments at the end of Section 2.4 of the 2nd edition of the quoted book by Albiac and Kalton. Though in Polish, there is also an adequate comment on page 284 of the quoted monograph of Alexiewicz, Orlicz’s first PhD-student, still in the occupied Lwów. In Grothendieck proved a theorem, whose special case is the Orlicz–Pettis theorem in locally convex spaces. Later, a more direct proofs of the form (i) of the theorem in the locally convex case were provided by McArthur and Robertson. Orlicz-Pettis type theorems. The theorem of Orlicz and Pettis had  been strengthened and generalized in many directions. An early survey of this area of research is Kalton's paper. A natural setting for subseries convergence is that of an Abelian topological group formula_4 and a representative result of this area of research is the following theorem, called by Kalton the Graves-Labuda-Pachl Theorem. Theorem. Let formula_4 be an Abelian group and formula_23 two Hausdorff group topologies on formula_4 such that formula_25 is sequentially complete, formula_26, and the identity formula_27 is universally measurable. Then the subseries convergence for both topologies formula_28 and formula_29 is the same. As a consequence, if  formula_25 is a sequentially complete K-analytic group, then the conclusion of the theorem is true for "every" Hausdorff group topology formula_31 weaker than formula_32. This is a generalization of an analogical result for a sequentially complete analytic group formula_33 (in the original statement of the Andersen-Christensen theorem the assumption of sequential completeness is missing), which in turn extends the corresponding theorem of Kalton for a Polish group, a theorem that triggered this series of papers. The limitations for this kind of results are provided by the weak* topology of the Banach space formula_34 and the examples of F -spaces formula_4 with separating dual formula_2 such that the weak (i.e., formula_37) subseries convergence does not imply the subseries convergence in the F-norm of the space formula_4.