Some remarks on rearrangements of series

A N N A LES SO C IETA T IS M A TH EM ATICAE PO LO N AE Series I : COM M ENTATIONES M ATH EM ATICAE X V I I I (1974) RO C ZN IK I PO LSK IEG O T O W A R Z Y S T W A M ATEM AT Y CZNE G O

Séria I : P R A C E M A TEM ATYCZN E X V I I I (1974)

L. D rewnowski (Poznan)

Some remarks on rearrangements of series

Let P be the set of all permutations (one-to-one mappings onto) of N = (1, 2, ...} , endowed with the topology of pointwise convergence on N. This topology is determined for example by the metric d(p, q)

oo

= £ 3 - t m in {l, \p{i) — q(i)\}- For every p e P and ne N, the open ball

г=1

B ( p ,3 ~ n) in (P, d) is equal to { q e P : q ( i ) = p (i) for 1 < i < n}. The metric space (P , d) is not complete, nevertheless

L emma 1 [1]. P is o f the second B aire category.

OO

P ro o f. Suppose that P = ( J P w, where P n are closed and boundary.

n ⁼¹

Then it is easy to define a sequence (p n) in P and an increasing sequence ( K ) in N so that B (Pn, 3~kn )n P n = 0 , P (p n+1, 3 “^+i) с B ( p n, 3~k») and p n(i) = n for some i < n€ N. Then p = limp n exists in P , p (i) = p n{i) for % ^ and p $ Р и; n e Af. ^This is impossible.

R. P. Agnew has proved in [1] that, roughly, if a scalar series converges, but not unconditionally, then the set of those p e P for which

converges is of the first category. This was recently generalized by Talaljan [5] to series in a real Hilbert space. The purpose of this note is to show that analogous results are valid in arbitrary topological groups or vector spaces.

L emma 2. Let F be any fam ily o f fin ite sequences i n N. Let P (F )

oo

= U p nj where P n = {ре P : (p(i), p ( i + l ) , . . . , p { j ) ) e F fo r j > г > n).

1

Then either P (F ) = P or P ( F ) is o f the first category.

P ro o f. Suppose that there is qe P \ P (F ). Then we can find sequences (4) and (jk) in N such that i k < j k < i k+l and (q(ik), . . . , q (jk) ) i F , he N.

We are going to show that each P n is nowhere dense. Take an arbitrary p 0€ P and m e N, m > n, and then choose h so large that min{#(i): ik ^ i

< j*} > т а х {р ⁰ (г): 1 < i < m ]. Then any permutation p such that p (i)

L. Drewnowski

= p 0(i) for 1 < г < т and p (i) = q(i) for ik ^ i ^ j k , is in B ( p 0,3 ~ m) but not in P n. Thus P n is boundary, and being evidently closed, it is nowhere dense.

Below, G denotes a topological group, X a topological vector space, and (x{)ieN is a sequence in G or X . Group operation is written additively.

T heorem 1 . Suppose that G is sequentially complete. Then the set P c

oo

o f those p a P fo r which £ xp^ converges is either identical with P or o f the f= i

first category.

P ro o f. Given a neighbourhood U of the origin in G, let F v denote

m

the set of all finite sequences (ix, . im) such that У xh a U. We apply

k = l k

Lemma 2 for F = F v , writting P (U ) instead of P ( F U). If, for some U, P (U ) is of the first category, then so is P c becauseP c c P {U ). Otherwise

oo

P (U ) = P for every neighbourhood U, thus xp{i) satisfies the Cauchy

i = i

condition for every p a P . I t follows that P c = P .

П

T heorem 2. Let (х{) a X and let P b = [pa P : the sequence ( xp^)neN is

i= 1

bounded}. Then either P b — P or P b is o f the first category.

P ro o f. There are two possibilities:

OO

1° For every neighbourhood U of 0, P = U P (nU ). Then by Lemma

n — l

2, P = P (n U ) for some n. I t follows that P b = P .

oo

2° There is U such that P Ф \^JP(nU). Then, by Lemma 2 again,

n = 1

the latter set is of the first category and contains P b.

P b — P means simply that the set of all finite sums x{i + ... + xik 00

is bounded. In this case the series ^ x{ is said to be perfectly bounded

i = l

([*2]? [3]). A linear topological space X is said to satisfy condition (0) if every perfectly bounded series of its points is convergent ; each such series is then subseriesly convergent, hence also unconditionally. Every weakly sequentially complete locally convex vector space, in particular every Hilbert space, satisfies condition (0) (this follows easily from the Orlicz- Pettis theorem). (See [2], [3] and [4] Chapter I I I , § ⁸ for other spaces satisfying this condition.) Therefore the results from [ 1 ] and [5] mentionned at the beginning are contained in the following corollary to Theorem 2.

oo

C orollary . Let X satisfy condition ( 0 ). I f a series xi in X is con-

г = 1

vergent but not unconditionally, then P b is o f the first category.

Rearrangements of series

R e m a rk . One obtains similar results (essentially due to Orlicz [3]) when instead of permutations of a series £ xi considers its subseries xk.

(&x< &2< . . . ) ; P is then replaced by éP{N), the power set of N.

References

[1] R. P . A g n e w ,

Bull. Amer. Math. Soc. 46 (1940), p. 7 9 7 -799.

[2] W . M a tu s z e w s k a and W . O r lic z ,

Bull. Acad.

Polon. Sci. 16 (1968), p. 8 0 1 -8 0 7 .

[3] W . O rlic z ,

[Polish;

English sum m ary], Prace M at. 1 (1955), p. 3 9 3 -414.

[4] S. R o le w ic z ,

PW N , W arszaw a 1972.

[5] F . A. T a l a l j a n ,

[Russian], Mat.

Zametki 12 (1972), p. 2 7 5 -2 8 0 .

IN S T IT U T E OF M ATHEM ATICS, A. M IC K IE W IC Z U N IV E R S IT Y , POZNAN