A note on unconditionally convergent series in a complete topological ring
aa r X i v : . [ m a t h . GN ] S e p A NOTE ON UNCONDITIONALLY CONVERGENT SERIES IN ACOMPLETE TOPOLOGICAL RING
ALEX RAVSKY
Abstract.
We answer a question concerning classes of complete topological rings whereunconditionally convergent series have a special property.
In this note we answer Question 1 posed by a user dch at Mathematics Stack Exchangea few months ago [3]. In order to formulate it, we introduce the following notion. Atopological ring R is a dch ring , if for any unconditionally convergent series P ∞ i =1 a i in R and any neighborhood S of the additive identity 0 of R there exists a neighborhood S ′ of0 such that for any finite subset F of S ′ and any sequence ( f i ) of elements of F , we havethat P ∞ i =1 f i a i converges to a point in S . Question 1.
Are there well-known classes (e.g. locally compact, locally connected, regu-lar) of complete Hausdorff topological rings which are dch rings?
We show that a topological ring which is a Banach space (and so a connected completemetric space) can fail to be a dch ring, see Proposition 4. On the other hand, a topologicalring R is a dch ring provided R is locally compact Hausdorff, see Proposition 7 or R hasa base at the zero consisting of open ideals and the additive topological group of R issequentially complete, see Proposition 3.All groups explicitly considered in this paper are supposed to be Abelian, add alltopological groups (including additive groups of topological rings) are Hausdorff. Anendomorphism of a group G is a homomorphism from G to itself. A series P ∞ i =1 a i in atopological group, is unconditionally convergent , if there exists an element a ∈ G suchthat for each permutation σ of N , a series P ∞ i =1 a σ ( i ) converges to a .A non-negative function k · k defined in a group G is called a norm , provided for each g, h ∈ G , k g k = 0 iff g = 0, k− g k = k g k , and k g + h k ≤ k g k + k h k . It is well known and easyto show that that in this case a function d such that d ( g, h ) = k g − h k for each g, h ∈ G ,is a metric on G and the group G endowed with the metric d is a topological group. Aseries P ∞ i =1 a i on this group ( G, d ) group is absolutely convergent , if a series P ∞ i =1 k a i k converges in R . It is well known and easy to show (see, for instance, [2, Exercise 1.36])that each absolutely convergent series on ( G, d ) is unconditionally convergent iff (
G, d ) iscomplete.Whether unconditional convergence implies absolute convergence is a more delicateissue. If G = R n with the standard Euclidean norm, then by the Riemann RearrangementTheorem unconditional convergence implies absolute convergence. On the other hand, byDvoretzky-Rogers theorem [4], in any infinite dimensional real Banach space there existsa series which is unconditionally convergent but not absolutely convergent. Mathematics Subject Classification.
Key words and phrases. topological ring, unconditional convergence, locally compact topological ring,locally compact Abelian topological group.
By Proposition 1.27 from [2], convergence in a complete, non-Archimedean normedgroup G is much simpler. If ( a i ) is a sequence in G then the series P ∞ i =1 a i is uncondi-tionally convergent iff it is convergent iff ( a i ) converges to the identity of G .A sequence ( x n ) on a topological group G is Cauchy if for every neighborhood U of theidentity of G there is natural n such that x i − x j ∈ U , for each i, j ≥ n . A topologicalgroup G is sequentially complete , if each Cauchy sequence on G converges. Lemma 2.
Let G be a sequentially complete topological group, P ∞ i =1 a i be an uncondi-tionally convergent in G series, and ( f i ) ∞ i =1 be a sequence of continuous endomorphismsof G such that a set F of values of ( f i ) is finite. Then a series P ∞ i =1 f i ( a i ) converges.Proof. Let f ∈ F be any element. For each natural i put x fi = P k ≤ i, f k = f a i (we putthat the sum over an empty set equals the identity of G ). It is easy to show that theunconditional convergence of the series P ∞ i =1 a i , implies that a sequence ( x fi ) is Cauchy.Since the group G is sequentially complete, the sequence ( x fi ) converges to a point x f ∈ G .Let n be any natural number. Since each map f ∈ F is a homomorphism, P ni =1 f i ( a i ) = P f ∈ F f ( x fn ). Since each map f ∈ F is continuous and the sequence ( x fi ) converges to x f ,the sequence (cid:16)P f ∈ F f ( x fn ) (cid:17) n ∈ N converges to P f ∈ F f ( x f ). (cid:3) Proposition 3.
Each topological ring R which has a base at the zero consisting of openideals, and whose additive topological group is sequentially complete, is a dch ring.Proof. Let P ∞ i =1 a i be any convergent series in R and S be any neighborhood of the zero0 of R . Let S ′ be an arbitrary open ideal contained in S . Since each open subgroupof a topological group is closed in it, S ′ is closed in R . Let ( f i ) be any sequence ofelements of S ′ and T be any open ideal in R . Since the sequence ( a i ) converges to 0,there exists N such that a n ∈ T for each n ≥ N . Since T is ideal, it follows that P ni = m f i a i ∈ T for each m, n ≥ N . Since R has a base at the zero consisting of openideals, ( P ni =1 f i a i ) is a Cauchy sequence. Since the additive group of R is sequentiallycomplete, the series P ∞ i =1 f i a i converges to a point a ∈ R . Since S ′ is an ideal, for anynatural n , P ni =1 f i a i ∈ S ′ , so we have a ∈ S ′ = S ′ ⊂ S . (cid:3) Let p ∈ [1 , ∞ ] be any number and R p be a ring of bounded linear operators on a normedspace ( ℓ p , k · k ) of sequences x = ( x n ) n ∈ N of real numbers such that k x k p = P | x n | p < ∞ for p < ∞ and k x k = sup | x n | < ∞ for p = ∞ , such that ( ab )( x ) = b ( a ( x )) for each a, b ∈ R p and x ∈ ℓ p . It is easy to check that R p endowed with a topology generated byan operator norm k a k = sup {k a ( x ) k : x ∈ ℓ p with k x k ≤ } is a topological ring and aBanach space. Proposition 4.
For any p ∈ [1 , ∞ ] , the ring R p is non-dch.Proof. For each i and each x = ( x n ) ∈ ℓ p let a i ( x ) has the only non-zero i -th coordinate,which equals x i /i . Then a series P ∞ i =1 a i unconditionally converges. Now let S be a unitball centered at the zero 0 of R p and S ′ be any open neighborhood of 0. There exists ε > S ′ contains a closed ball of radius ε centered at 0. For each i and each x = ( x n ) ∈ ℓ p let f ′ i ( x ) has the only non-zero n -th coordinate, which equals εx . Since theharmonic series P ∞ n =1 /n diverges, there exists N such that ε P Nn =1 /n >
1. Put f i = f ′ i ,if i ≤ N and f i = 0, otherwise. Then f ′ i ∈ S ′ for each i . Let x = (1 , , , . . . ) ∈ ℓ p . Then N UNCONDITIONALLY CONVERGENT SERIES IN A COMPLETE TOPOLOGICAL RING 3 k x k p = 1, but (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X i =0 f i a i ! ( x ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = ε N X n =1 /n > . So P ∞ i =0 f i a i S . (cid:3) Let X be a non-empty set and ℓ ∞ ( X ) be a linear space of bounded real-valued functionson X endowed with the supremum norm k a k = sup x ∈ X | a ( x ) | for each a ∈ ℓ ∞ ( X ). Then X is a Banach space and a topological ring with respect to the coordinatewise additionand multiplication. Proposition 5.
For each non-empty set X , R = ℓ ∞ ( X ) is a dch ring.Proof. Let P ∞ i =1 a i be any unconditionally convergent series in R and S be any neighbor-hood of 0 of R . There exists ε > ∈ B ǫ = { b ∈ R : k b k ≤ ε } ⊂ S . Since foreach x ∈ X a series P ∞ i =1 a i ( x ) is unconditionally convergent, it is absolutely convergent,that is a ( x ) = P ∞ i =1 | a i ( x ) | < ∞ . If A = sup x ∈ X a ( x ) = ∞ then we can inductivelyconstruct a sequence ( x n ) of elements of X and an increasing sequence ( m n ) of naturalnumbers and a permutation σ of N such that a sequence ( (cid:12)(cid:12)P m n i =1 a σ ( i ) ( x n ) (cid:12)(cid:12) ) is unbounded,which contradicts the unconditional convergence of P ∞ i =1 a i . Pick δ > δA ≤ ε .Then S ′ = B δ = { b ∈ R : k b k ≤ δ } is a neighborhood of 0 of R . Let ( f i ) be any sequenceof elements of S ′ such that a set of values of ( f i ) is finite. By Lemma 2, a series P ∞ i =1 f i a i converges to some point a ∈ R . Since for each x ∈ X and each natural n we have (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 f i ( x ) a i ( x ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ n X i =1 k f i ( x ) kk a i ( x ) k ≤ n X i =1 δ k a i ( x ) k ≤ δ ∞ X i =1 k a i ( x ) k ≤ δA ≤ ε, we see that P ∞ i =1 f i a i ∈ B ǫ and thus ∈ B ǫ = B ǫ ⊂ S . (cid:3) Recall that a character of a topological group G is a continuous homomorphism from G to a group T = { z ∈ C : | z | = 1 } , endowed with the multiplication and topology inheritedfrom C . Lemma 6.
Let G be a locally compact topological group. Let E be a set of endomor-phisms of G containing the zero endomorphism E and endowed with a topology such thatthe evaluation map from E × G to G , ( f, x ) f ( x ) for each f ∈ E and x ∈ G , iscontinuous. Let a series P ∞ i =1 a i of elements a i of G unconditionally converges. Then forany neighborhood S of the identity of G there exists an open neighborhood S ′ of E suchthat for any sequence ( f i ) of elements of S ′ and any natural n , P ni =1 f i ( a i ) ⊂ S .Proof. Shrinking the neighborhood S , if needed, we can assume that its closure S iscompact. Since the series P ∞ i =1 a i converges, all but finitely many a i ’s belong to S . Thusa set S ∪ { a i } is compact. Let G ′ be the smallest subgroup of G containing S ∪ { a i } .Since G ′ ⊃ S , G ′ is an open subgroup of G . By Example 69 from [5], the group G ′ is topologically isomorphic to a product H × Z p × R q , where H is a compact group, Z and R are the groups of integer and real numbers, respectively, endowed with the usualtopologies, and p, q are non-negative integers. By [5, § x of H there exists a character χ from H to T such that χ ( x ) = 0. It follows that H can betopologically isomorphically embedded into a power T κ for some cardinal κ . So thereexists an isomorphic topological embedding i : G ′ → T κ × R s for a non-negative integer s = p + q . ALEX RAVSKY
Thus there exist 0 < ε ≤ π and a finite set Γ of continuous homomorphisms (which arecompositions of i and projections on the factors) from G ′ to a group K (which is R or T )such that S ⊃ S = { x ∈ G ′ : k γ ( x ) k ≤ ε for each γ ∈ Γ } , where k g k = | g | , if g ∈ R and k g k = | arg g | ≤ π , if g ∈ T . Remark that for each g, h ∈ K we have k g + h k ≤ k g k + k h k and if k g k ≤ π/ k g + g k = 2 k g k . For each x ∈ G ′ put k x k Γ = max γ ∈ Γ k γ ( x ) k .Let γ be any element of Γ. Since the series P ∞ i =1 a i unconditionally converges and γ iscontinuous, a series P ∞ i =1 γ ( a i ) unconditionally converges too. This easily follows that aseries P ∞ i =1 k γ ( a i ) k converges. Since the set Γ is finite, M = 1 + 1 ε ∞ X i =1 k a i k Γ ≤ ε ∞ X i =1 X γ ∈ Γ k γ ( a i ) k < ∞ . Pick 0 < δ ≤ ε/ (2 M ).Since S is a closed subset of a compact space S , it is compact. Let x ∈ S be anyelement. Since 0 = k E ( x ) k Γ < δ and both the evaluation map from E × G to G and k · k Γ are continuous, there exist a neighborhood S ′′ x of 0 E and an open neighborhood O x of x in S such that k g k Γ < δ for each g ∈ S ′′ x ( O x ), where S ′′ x ( O x ) = { f ( y ) : f ∈ S ′′ x , y ∈ O x } .Since the set S is compact, there exists a finite subset F of S such that S ⊂ S { O x : x ∈ F } . Put S ′′ = T { S ′′ x : x ∈ F } . It follows k f ( x ) k Γ < δ for each f ∈ S ′′ and x ∈ S .Suppose to the contrary that there exist f ∈ S ′′ and x ∈ S such that M k f ( x ) k Γ > k x k Γ .Since 0 < k f ( x ) k Γ < δ , there exists the largest natural k such that 2 k k f ( x ) k Γ < δ ≤ ε ≤ π . Then 2 k k f ( x ) k Γ ≥ δ . Let 2 k x be a sum of 2 k instances of x . It is easy to see that ε ≥ M δ > M k f (2 k x ) k Γ = 2 k M k f ( x ) k Γ > k k x k Γ . It follows 2 k x ∈ S and so k f (2 k x ) k Γ < δ , a contradiction.Since the series P ∞ i =1 a i converges, a set { a i } \ S is finite. Since k E ( x ) k Γ = 0 for each x ∈ { a i } \ S and both the evaluation map from E × G to G and k · k Γ are continuous,there exists an open neighborhood S ′ ⊂ S ′′ of 0 E such that k f ( x ) k Γ < δ < k x k Γ /M foreach f ∈ S ′ and each x ∈ { a i } \ S . Thus we have k f ( a i ) k Γ ≤ k a i k Γ /M for each f ∈ S ′ each a i .Let ( f i ) be any sequence of elements of S ′ , x = P ni =1 f i ( a i ). Then f i ( a i ) ∈ G ′ for each i , so x ∈ G ′ and k x k Γ ≤ n X i =1 k f i ( a i ) k Γ ≤ n X i =1 k a i k Γ /M ≤ ε. Thus x ∈ S ⊂ S . (cid:3) Proposition 7.
Each locally compact Hausdorff topological ring R is dch.Proof. Let P ∞ i =1 a i be any unconditionally convergent series in R and S be any neighbor-hood of the zero 0 of R . It is well known that each topological group is regular, so itsuffices to show that there exists a neighborhood S ′ of 0 such that for any finite subset F of S ′ and any sequence ( f i ) of elements of F , we have that P ∞ i =1 f i a i converges to a pointin S . Since for each f ∈ R a map R → R , x f x for each x ∈ R is a continuous homo-morphism of the additive group of R , Lemma 6 implies that there exists a neighborhood S ′ of 0 such that for any sequence ( f i ) of elements of S ′ and any natural n , P ni =1 f i a i ∈ S .Since the group G is locally compact, it is sequentially complete, for instance, by Theorem3.6.24 from [1], by Lemma 2, the series P ni =1 f i a i converges to a point a ∈ R . Since forany natural n , P ni =1 f i a i ∈ S , we have a ∈ S . (cid:3) N UNCONDITIONALLY CONVERGENT SERIES IN A COMPLETE TOPOLOGICAL RING 5
Acknowledgment.
The author thanks to Taras Banakh for the careful reading of thepaper manuscript and its helpful discussion.
References [1] Alexander Arhangel’ski˘ı, Mikhail Tkachenko,
Topological Groups and Related Structures , AtlantisPress, 2008.[2] Pete L. Clark,
Algebraic Number Theory II: Valuations, Local Fields and Adeles , math.uga.edu/~pete/8410FULL.pdf .[3] dch, Unconditional convergence of a sum of elements in a complete Hausdorff topological ring , math.stackexchange.com/q/3702472 .[4] A. Dvoretzky, C.A. Rogers, Absolute and unconditional convergence in normed linear spaces , Proc.Nat. Acad. Sci. USA (1950), 192–197.[5] L.S. Pontrjagin, Continuous groups , 2nd ed., M., 1954 (in Russian).
Alex Ravsky: Pidstryhach Institute for Applied Problems of Mechanics and Mathe-matics National Academy of Sciences of Ukraine
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