Permutations, Signs, and Sum Ranges

The sum range SRx;X, for a sequence x=(xn)n∈N of elements of a topological vector space X, is defined as the set of all elements s∈X for which there exists a bijection (=permutation) π:N→N, such that the sequence of partial sums (∑k=1nxπ(k))n∈N converges to s. The sum range problem consists of describing the structure of the sum ranges for certain classes of sequences. We present a survey of the results related to the sum range problem in finite- and infinite-dimensional cases. First, we provide the basic terminology. Next, we devote attention to the one-dimensional case, i.e., to the Riemann–Dini theorem. Then, we deal with spaces where the sum ranges are closed affine for all sequences, and we include some counterexamples. Next, we present a complete exposition of all the known results for general spaces, where the sum ranges are closed affine for sequences satisfying some additional conditions. Finally, we formulate two open questions.

[1]  Vaja Tarieladze,et al.  Series with Commuting Terms in Topologized Semigroups , 2021, Axioms.

[2]  G. Giorgobiani.  Rearrangements of Series , 2019, Journal of Mathematical Sciences.

[3]  J. O. Wojtaszczyk.  A series whose sum range is an arbitrary finite set , 2008, 0803.0415.

[4]  M. Talagrand.  Type and infratype in symmetric sequence spaces , 2004 .

[5]  V. Tarieladze,et al.  Nuclear and GP-Nuclear Groups , 2000 .

[6]  A. Defant,et al.  The Levy-Steinitz rearrangement theorem for duals of metrizable spaces , 1999, math/9908112.

[7]  M. Talagrand.  Type, infratype and the Elton-Pajor theorem , 1992 .

[8]  P. Kornilov.  ON THE SET OF SUMS OF A CONDITIONALLY CONVERGENT SERIES OF FUNCTIONS , 1990 .

[9]  D. V. Pecherskiĭ.  REARRANGEMENTS OF SERIES IN BANACH SPACES AND ARRANGEMENTS OF SIGNS , 1989 .

[10]  N. Vakhania,et al.  Probability Distributions on Banach Spaces , 1987 .

[11]  S. Chobanyan.  STRUCTURE OF THE SET OF SUMS OF A CONDITIONALLY CONVERGENT SERIES IN A NORMED SPACE , 1987 .

[12]  P. Kornilov.  ON REARRANGEMENTS OF CONDITIONALLY CONVERGENT SERIES OF FUNCTIONS , 1982 .

[13]  D. V. Pečerskiĭ.  a Theorem on Projections of Rearranged Series with Terms in L_p , 1977 .

[14]  Y. Katznelson,et al.  Conditionally convergent series in $R^{\infty}$. , 1974 .

[15]  E. M. Nikishin.  Rearrangements of series in Lp , 1973 .

[16]  B. Kashin.  On a property of functional series , 1972 .

[17]  V. Fonf.  Conditionally convergent series in a uniformly smooth Banach space , 1972 .

[18]  E. M. Nikišin.  REARRANGEMENTS OF FUNCTION SERIES , 1971 .

[19]  E. M. Nikishin.  On the set of sums of a functional series , 1970 .

[20]  R. Dudley.  Corrections to: “On sequential convergence” , 1970 .

[21]  V. Drobot.  Rearrangements of series of functions , 1969 .

[22]  R. Dudley.  ON SEQUENTIAL CONVERGENCE , 1964 .

[23]  H. Hadwiger.  Über die Konvergenzarten unendlicher Reihen im Hilbertschen Raum , 1942 .

[24]  H. Hadwiger.  Über das Umordnungsproblem im Hilbertschen Raum , 1940 .

[25]  W. Gross,et al.  Bedingt konvergente Reihen , 1917 .

[26]  Ulisse Dini.  Sui prodotti infiniti , 1868 .

[27]  Paul B. Garrett.  Topological vector spaces , 2016 .

[28]  S. Levental,et al.  On rearrangement theorems in Banach spaces , 2014 .

[29]  L. Maligranda.  Józef Marcinkiewicz (1910-1940) - on the centenary of his birth , 2011 .

[30]  M. A. Sofi,et al.  LEVY - STEINITZ THEOREM IN INFINITE DIMENSION , 2008 .

[31]  P. Rosenthal.  The Remarkable Theorem of Levy and Steinitz , 2007 .

[32]  M. J. Chasco,et al.  On rearrangements of series in locally convex spaces. , 1997 .

[33]  S. Chobanyan,et al.  Convergence A.S. of Rearranged Random Series in Banach Space and Associated Inequalities , 1994 .

[34]  W. Banaszczyk.  Rearrangement of series in nonnuclear spaces , 1993 .

[35]  W. Banaszczyk.  Balancing vectors and convex bodies , 1993 .

[36]  Wojciech Banaszczyk,et al.  The Steinitz theorem on rearrangement of series for nuclear spaces. , 1990 .

[37]  V. Drobot.  A note on rearrangements of series , 1970 .

[38]  C. Mcarthur.  On Relationships Amongst Certain Spaces Of Sequences In An Arbitrary Banach Space , 1956, Canadian Journal of Mathematics.

[39]  A. Dvoretzky,et al.  Convergence- and sum-factors for series of complex numbers , 1951 .

[40]  E. Steinitz.  Bedingt konvergente Reihen und konvexe Systeme. , 1913 .

[41]  P. Levy,et al.  Sur les séries semi-convergentes , 2022 .

[42]  M. Fréchet.  Sur le résultat du changement de l'ordre des termes dans une série , 1903 .