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A more thorough discussion of the origins of the Orlicz-Pettis theorem and, in particular, of the paper  ^[1] can be found in ^[2]. See also footnote 5 on p. 839 of  ^[3]  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 ^[4] , still in the occupied Lwów.

In ^[5] 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. ^[6] ^[7]

Orlicz-Pettis type theorems

The theorem of Orlicz and Pettis had been strengthened and generalized in many directions. An early survey is Kalton's paper. ^[8] A natural setting for subseries convergence is that of an Abelian topological group $X$ and a representative result of this area of research is the following theorem, called by Kalton the Graves-Labuda-Pachl Theorem. ^[9] ^[10] ^[11]

Theorem. Let $X$ be an Abelian group and $\alpha ,\beta$ two Hausdorff group topologies on $X$ such that $(X,\beta )$ is sequentially complete, $\alpha \subset \beta$ , and the identity $j:(X,\alpha )\to (X,\beta )$ is universally measurable. Then the subseries convergence for both topologies $\alpha$ and $\beta$ is the same.

As a consequence, if $(X,\beta )$ is a sequentially complete K-analytic group, then the conclusion of the theorem is true for every Hausdorff group topology $\alpha$ which is weaker than $\beta$ . This is a generalization of an analogical result for a sequentially complete analytic group $(X,\beta )$ ^[12] (in the original statement of the Andersen-Christen theorem the assumption of sequential completeness is missing ^[13]), which in turn extends the corresponding theorem of Kalton for Polish group ^[14], a theorem that triggered this series of papers.

The limitations for this kind of results are provided by the wak* topology of the Banach space $\ell ^{\infty }$ and the examples of F-spaces $X$ with separating dual $X^{*}$ such that the weak (i.e., $\sigma (X,X^{*})$ ) subseries convergence does not imply the subseries convergence in the F-norm of the space $X$ . ^[15] ^[16]

References

^ W. Orlicz, Beiträge zur Theorie der Orthogonalentwicklungen II, Studia Math. 1 (1929), 241–255.
^ W. Filter and I. Labuda, Essays on the Orlicz-Petts theorem, I (The two theorems), Real Anal. Exchange 16(2), 1990-91, 393--403.
^ W. Orlicz, Collected works, Vol.1, PWN-Polish Scientific Publishers, Warszawa 1988.
^ https://www.genealogy.math.ndsu.nodak.edu/id.php?id=51907&fChrono=1
^ A.Grothendieck, Sur les applications linéaires faiblement compacts d'espaces du type C(K), Canadian J. Math 3 (1953), 129--173.
^ C.W. McArthur On a theorem of Orlicz and Pettis, Pacific J. Math 22 (1967), 297--302.
^ A.P. Robertson, On unconditional convergence in topological vector spaces, Proc. Roy. Soc. Edinburgh A, 68 (1969), 145--157.
^ Nigel Kalton, The Orlicz-Pettis theorem, Contemporary Mathematics 2 (1980), 91–100.
^ I. Labuda, , [1] Universal measurability and summable families in topological vector spaces, Indag. Math. (N.S.) 82(1979), 27-34.
^ J. K. Pachl, A note on the Orlicz-Pettis Theorem,[2] Indag. Math. (N.S.)82 (1979), 35-37.
^ W. H. Graves, [3] Universal Lusin measurability and subfamily summable families in Abelian topological groups, Proc. Amer. Math. Soc. 73 (1979), 45--50.
^ N. J. M. Andersen and J. P. R. Christensen, Some results on Borel structures with applications to subseries convergence in Abelian topological groups, Israel J. Math. 15 (1973), 414--420.
^ I. Labuda [4], Measure, Category and Convergent Series, Real Anal. Exchange 32(2) (2017), 411--428.
^ N. J. Kalton, [5] Subseries convergence in topological groups and vector measures, Israel J. Math. 10 (1971), 402-412.
^ M. Nawrocki, [6] On the Orlicz-Pettis property in non-locally convex F-spaces, Proc. Amer. Math. Soc. 101(1987), 492--–496.
^ M. Nawrocki, [7] The Orlicz-Pettis theorem fails for Lumer's Hardy spaces $(LH)^{p}(B)$ , Proc. Amer. Math. Soc. 109 (1990), 957–963.

Alexiewicz, Andrzej (1969). Analiza Funkcjonalna. Państwowe Wydawnictwo Naukowe, Warszawa..
Albiac, Fernando; Kalton, Nigel (2016). Topics in Banach space theory, 2nd ed. Springer. ISBN 9783319315553..

Category:Theorems in functional analysis

[1] W. Orlicz, Beiträge zur Theorie der Orthogonalentwicklungen II, Studia Math. 1 (1929), 241–255.

[2] W. Filter and I. Labuda, Essays on the Orlicz-Petts theorem, I (The two theorems), Real Anal. Exchange 16(2), 1990-91, 393--403.

[3] W. Orlicz, Collected works, Vol.1, PWN-Polish Scientific Publishers, Warszawa 1988.

[4] ttps://www.genealogy.math.ndsu.nodak.edu/id.php?id=51907&fChrono=1

[5] A.Grothendieck, Sur les applications linéaires faiblement compacts d'espaces du type C(K), Canadian J. Math 3 (1953), 129--173.

[6] C.W. McArthur On a theorem of Orlicz and Pettis, Pacific J. Math 22 (1967), 297--302.

[7] A.P. Robertson, On unconditional convergence in topological vector spaces, Proc. Roy. Soc. Edinburgh A, 68 (1969), 145--157.

[8] Nigel Kalton, The Orlicz-Pettis theorem, Contemporary Mathematics 2 (1980), 91–100.

[9] I. Labuda, , [1] Universal measurability and summable families in topological vector spaces, Indag. Math. (N.S.) 82(1979), 27-34.

[10] J. K. Pachl, A note on the Orlicz-Pettis Theorem,[2] Indag. Math. (N.S.)82 (1979), 35-37.

[11] W. H. Graves, [3] Universal Lusin measurability and subfamily summable families in Abelian topological groups, Proc. Amer. Math. Soc. 73 (1979), 45--50.

[12] N. J. M. Andersen and J. P. R. Christensen, Some results on Borel structures with applications to subseries convergence in Abelian topological groups, Israel J. Math. 15 (1973), 414--420.

[13] I. Labuda [4], Measure, Category and Convergent Series, Real Anal. Exchange 32(2) (2017), 411--428.

[14] N. J. Kalton, [5] Subseries convergence in topological groups and vector measures, Israel J. Math. 10 (1971), 402-412.

[15] M. Nawrocki, [6] On the Orlicz-Pettis property in non-locally convex F-spaces, Proc. Amer. Math. Soc. 101(1987), 492--–496.

[16] M. Nawrocki, [7] The Orlicz-Pettis theorem fails for Lumer's Hardy spaces $(LH)^{p}(B)$ , Proc. Amer. Math. Soc. 109 (1990), 957–963.

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