View source for Subseries convergence

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120290/s1202901.png" /> is a Hausdorff Abelian [[Topological group|topological group]], a series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120290/s1202902.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120290/s1202903.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120290/s1202905.png" />-subseries convergent (respectively, unconditionally convergent) if for each subsequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120290/s1202906.png" /> (respectively, each permutation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120290/s1202907.png" />) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120290/s1202908.png" />, the subseries <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120290/s1202909.png" /> (respectively, the rearrangement <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120290/s12029010.png" />) is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120290/s12029011.png" />-convergent in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120290/s12029012.png" />. In one of the early papers in the history of [[Functional analysis|functional analysis]], W. Orlicz showed that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120290/s12029013.png" /> is a weakly sequentially complete [[Banach space|Banach space]], then a series in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120290/s12029014.png" /> is weakly unconditionally convergent if and only if the series is norm unconditionally convergent [[#References|[a5]]]. Later, he noted that if  "unconditional convergence"  is replaced by  "subseries convergence" , the proof showed that the weak sequential completeness assumption could be dropped. That is, a series in a Banach space is weakly subseries convergent if and only if the series is norm subseries convergent; this result was announced in [[#References|[a1]]], but no proof was given. In treating some problems in vector-valued measure and integration theory, B.J. Pettis needed to use this result but noted that no proof was supplied and then proceeded to give a proof ([[#References|[a6]]]; the proof is very similar to that of Orlicz). The result subsequently came to be known as the Orlicz–Pettis theorem (see [[#References|[a3]]] for a historical discussion).

Since the Orlicz–Pettis theorem has many applications, particularly to the area of vector-valued measure and integration theory, there have been attempts to generalize the theorem in several directions. For example, A. Grothendieck remarked that the result held for locally convex spaces and a proof was supplied by C.W. McArthur. Recent (1998) results have attempted to push subseries convergence to topologies on the space which are stronger than the original topology (for references to these results, see the historical survey of [[#References|[a4]]]).

In the case of a [[Banach space|Banach space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120290/s12029015.png" />, attempts have been made to replace the [[Weak topology|weak topology]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120290/s12029016.png" /> by a weaker topology, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120290/s12029017.png" />, generated by a subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120290/s12029018.png" /> of the dual space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120290/s12029019.png" /> which separates the points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120290/s12029020.png" />. Perhaps the best result in this direction is the Diestel–Faires theorem, which states that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120290/s12029021.png" /> contains no subspace isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120290/s12029022.png" />, then a series in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120290/s12029023.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120290/s12029024.png" /> subseries convergent if and only if the series is norm subseries convergent. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120290/s12029025.png" /> is the dual of a Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120290/s12029026.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120290/s12029027.png" />, then the converse also holds (see [[#References|[a2]]], for references and further results).

J. Stiles gave what is probably the first extension of the Orlicz–Pettis theorem to non-locally convex spaces; namely, he established a version of the theorem for a complete metric linear space with a Schauder basis. This leads to a very general form of the theorem by N. Kalton in the context of Abelian topological groups (see [[#References|[a4]]] for references on these and further results).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Banach,   "Théoriè des opérations linéaires" , Monogr. Mat. Warsaw  (1932)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Diestel,   J. Uhl,   "Vector measures" , ''Surveys'' , '''15''' , Amer. Math. Soc.  (1977)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  W. Filter,   I. Labuda,   "Essays on the Orlicz–Pettis theorem I"  ''Real Anal. Exch.'' , '''16'''  (1990/91)  pp. 393–403</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  N. Kalton,   "The Orlicz–Pettis theorem"  ''Contemp. Math.'' , '''2'''  (1980)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  W. Orlicz,   "Beiträge zur Theorie der Orthogonalent wicklungen II"  ''Studia Math.'' , '''1'''  (1929)  pp. 241–255</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  B.J. Pettis,   "On integration in vector spaces"  ''Trans. Amer. Math. Soc.'' , '''44'''  (1938)  pp. 277–304</TD></TR></table>