A theorem on summable families in normed groups. Dedicated to the professors of mathematics. L. Berg, W. Engel, G. Pazderski, and H.- W. Stolle.
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1 Rostock. Math. Kolloq. 49, 51{56 (1995) Subject Classication (AMS) 46B15, 54A20, 54E15 Harry Poppe A theorem on summable families in normed groups Dedicated to the professors of mathematics L. Berg, W. Engel, G. Pazderski, and H.- W. Stolle. ABSTRACT. In functional analysis the notion of a summable family (with sum x) is wellknown. If (x i ) i2i is a family of points from a normed space, (x i ) is called summable to x i for each " > 0 there exits a nite set F 0 I such that kx? P i2f x i k < " for each nite set F, F 0 F I. But we can interprete this denition as the convergence of a suitable net. In a normed commutative group we characterize the fact that this net is a Cauchy net. KEY WORDS. Commutative topological group, summable family, special nets, normed (commutative) group, Cauchy nets 1 Introduction We consider a normed space (X; k k) and let (x i ) i2i be a family of points from X; x 2 X; the notion that \ (x i ) i2i is a summable family with the sum x " is well{known; one denes: 1.1 For each " > 0 there exists a nite set J 0 I such that kx? P i2j x i k < " for each nite J I, J 0 J. See for instance [2], [3], [5], [7], [8]. One can prove a \Cauchy criterion", and probably at rst in Hilbert spaces it has been found that for a summable family (x i ) i2i (with sum x) for all but countably many i 2 I we have x i = 0. But one can observe that denition 1.1 means that the following net (Moore{ Smith{sequence) converges to x with respect to the usual notion of convergence of nets in a topological space: let F = F (I) be the collection of all nite nonempty subsets of I and let be x F := P i2f x i where F 2 F. If F is ordered by inclusion then F = (F; ) is a poset which is directed: for F 0, F 00 2 F, F 0 [ F 00 2 F and F 0 [ F 00 is an upper bound for F 0, F 00. Hence F can serve as an index set for a net and the desired net is (x F ) F 2F. And of course \x F! x in (X; k k)" is equivalent to denition 1.1. Clearly, the limit point x is unique since (X; k k)
2 52 H. Poppe is a Hausdor topological space. In [2] and [8] this net is mentioned but from this net the corresponding lter is constructed and then this lter is used. But in my opinion it is more intiutive to use directly the net (x F ) F 2F. Our aim in this short note is rst to emphasize that we should distinguish between the notion that the net (x F ) F 2F converges and the notion that this net is a Cauchy net and second to establish the precise relationship between the convergence of the net (x F ) F 2F, the situation when (x F ) is a Cauchy net and the fact that most members of the family (x i ) i2i will vanish. The answer is given by the theorem (in section 3) and its corollary. In a second corollary we consider a short application to orthonormal families in an inner product space. Of course concerning the concrete subject of the paper both the assertions and the proofs are in some sense elementary. 2 Summable families of points in a commutative topological group Let us start with a set X and a family (x i ) i2i of points of X; we consider the index set F = F (I) as dened in section 1. If (X; +) is a group we can dene x F = P x i for each F ; i2f if (X; +) is commutative then x F is uniquely dened and (x F ) F 2F is a net in (X; +). Now let (X; +) be a commutative topological group and we denote by N(0) the neighbourhood lter of the zeroelement 0 2 X. Then V := fv U j U 2 N(0)g, where V U = f(x; y) 2 X Xj x? y 2 Ug, is a base of a natural (diagonal) uniformity for X. Hence we can use Cauchy nets in X. Let us briey recall this notion: If (X; V ) is an arbitrary uniform space with diagonal structure V then a net (x i ) i2i from X is called Cauchy net i for each V 2 V there exists i V 2 I such that i 1 ; i 2 i V implies (x i1 ; x i2 ) 2 V. But since for V the triangle inequality holds we also can dene a Cauchy net by: For V 2 V we nd i V 2 I: for each i i V holds: (x i ; x iv ) 2 V. Cauchy nets are used for instance in [4], [6]. For the proof of the theorem which we want to state in section 3 we need some lemmas. For the statements of the lemmas we assume that (X; +) is a commutative topological group, (x i ) i2i is a family from X and F is dened as above. The main technical argument in the proofs of the lemmas is the simple fact: if F 1, F 2 I and F 1 \ F 2 = ; then x F1 [F 2 = x F1 + x F2. For this reason we prove only the rst lemma (as an example). Lemma 1 Equivalent are: (1) (x F ) F 2F is a Cauchy net. (2) For each V 2 V there exists F 0 2 F such that for all F 2 F, F \ F 0 = ; implies (x F ; 0) 2 V.
3 A theorem on summable families in normed groups 53 Proof: (1) =) (2): Let V = V U, U 2 N(0), be a basic element from V ; then by (1) we nd F 0 2 F such that F 2 F and F 0 F imply (x F ; x F0 ) 2 V and hence x F? x F0 2 U; now let H 2 F and H \ F 0 = ;; then since F 0 F 0 [ H we get: x F0 [H? x F0 = (x F0 + x H )? x F0 = x H 2 U and thus (x H ; 0) 2 V. (2) =) (1): Let V = V U 2 V ; by (2) we nd F 0 2 F such that (x F ; 0) 2 V for each F 2 F, F \ F 0 = ;; now we consider an arbitrary H 2 F such that F 0 H; we can assume F 0 6= H; F 0 \ (HnF 0 ) = ; implies (x HnF0 ; 0) 2 V and hence x HnF0? 0 2 U; now x H? x F0 = x HnF0 + x F0? x F0 = x HnF0 2 U and hence (x H ; x F0 ) 2 V showing that (x F ) F 2F is a Cauchy net. Remarks: 1. This result is also proved in [2], [8], where the authors used the (Cauchy) lter corresponding to (x F ). 2. Since i =2 F 0 means fig \ F 0 = ; from 1 follows that if (x F ) is a Cauchy net then for each neighbourhood U of 0 there exists F 2 F such that x i 2 U for each i 2 InF. 3. In [1] families which fulll assertion (2) of lemma 1 are called summable, that means, (x i ) i2i is summable by the denition of the author i the net (x F ) F 2F is a Cauchy net. 4. We call (x i ) a Cauchy family if (x F ) is a Cauchy net. Lemma 2 Let A be a nonempty subset of I such that A 6= I and x i = 0 for each i 2 InA; let A = fb A j B nite g, hence A F. If (x F ) F 2A is a Cauchy net then (x F ) F 2F is a Cauchy net, too. Lemma 3 Let I be innite; let A I such that A 6= ;, A 6= I and x i = 0 for each i 2 InA; let A be dened as in lemma 2. If (x F ) F 2F converges in X, x F! x, then (x F ) F 2A! x. 3 Characterizing the fact that (x F ) F 2F is a Cauchy net Here we want to consider normed groups. We dene (see [8]): 3.1 Let (X; +) be a group; we call k k : X! R a norm for (X; +) i the axioms hold: 1. kxk 0 for each x 2 X; 2. kxk = 0 i x = 0;
4 54 H. Poppe 3. k? xk = kxk; 4. kx + yk kxk + kyk. Then as is it easy to see, d : d(x; y) = kx? yk denes a metric for X. Lemma 4 (X; +; k k) = (X; +; d) is a topological group. Proof: Each translation x! x + a is continuous and hence the topological space X is homogenous; now let be f : X! X, f(x) =?x; then if (x i ) is a net from X and x i! 0 we get: kf(x i )? f(0)k = k? x i? 0k = k? x i k = kx i k = kx i? 0k showing that f is continuous; since convergence in X X means coordinatewise convergence and by axiom 4. we see that the map (x; y)! x + y is continuous, too. Now we want to state the main result of our paper. Theorem 5 Let (X; +; k k) be a commutative normed group; let (x i ) i2i be a family of elements of X and let I be innite. We consider the net (x F ) F 2F, where F = F (I). Then the following statements are equivalent: (1) (x F ) is a Cauchy net. (2) (a) There exists a countable set I I such that x i = 0 for each i 2 InI. (b) (x F ) F 2I is a Cauchy net. Proof: By lemma 4 we see that we can use the lemmas of section 2 for the proof. (1) =) (2): By lemma 1 we get: for each n 2 N, n 1 there exists F n 2 F such that d(x F ; 0) = kx F k < 1 n for each F 2 F, F \ F S n = ;; hence: i 2 T n 1 F n implies fig \ F n = ; for each n and thus kx i k = kx fig k < 1 n for each n showing that x i = 0 holds. I = 1 S n=1 F n n=1 is countable and so (a) holds. To prove also (b) let " > 0 be given. We assume that I 6= I, otherwise there is nothing to prove. Since (x F ) is a Cauchy net we nd F " 2 F : kx F? x F" k < " for each F 2 F, F " F ; now F " \ I = ; or F " \ I 6= ;; if F " \ I = ; let F 1 2 I be xed and H 2 I, F 1 H; then F " F 1 [ F " H [ F " and hence: kx H? x F1 k = kx H[F"? x F1 [F " k = kx H[F"? x F" + x F"? x F1 [F " k < 2"; if F " \ I 6= ; let be H 2 I and F " \ I H; then F " H [ F " ni implies: kx H? x F"\I k = kx H[(F "ni )? x (F"\I )[(F "ni )k = kx H[(F"nI )? F " k < ". (2) =) (1): Setting A = I in lemma 2 we get the proof. Corollary 6 Under the assumptions of the theorem it holds: If the net (x F ) F 2F converges in X, x F! x, then we get:
5 A theorem on summable families in normed groups 55 (a) There exists a countable set I I such that x i = 0 for each i 2 InI : (b) (x F ) F 2I! x (c) For each! enumeration of I, I = fi 0 ; i 1 ; : : : g holds that the sequence kp x ij = (x fi0 ;i 1 ;::: ;i k g) k2n converges to x. j=0 k2n Proof: x F! x implies that (x F ) is a Cauchy net and hence we get (a) by theorem 1; then we get assertion (b) by lemma 3 if we substitute A = I in the lemma. If I = fi 0 ; i 1 ; : : : g and F k = fi 0 ; : : : ; i k g for each k 2 N then (ff k g; ) clearly is a conal subset of I and hence we get (c). Remark: Assertion (a) of corollary 6 is proved in [2], [8] and is stated (without proof) in [3]. Corollary 7 Let (X; (; )) be an inner product space and (x i ) i2i an arbitrary orthonormal family in X; let x be an arbitrary point in X; then there exists a countable set I I such that (x; x i ) = 0 for each i 2 InI. Proof: In [7] it is shown that ((x; x i )x i ) i2i is a Cauchy family; since (X; +; kk) is a commutative normed group by theorem 1 we nd a countable I I such that (x; x i )x i = 0 for each i 2 InI ; but x i being a member of an orthonormal family yields x i 6= 0 and hence (x; x i ) = 0. Remark: This result one nds in [5] (with another proof). References [1] Banaszczyk, W. : Summable families in nuclear groups. Studia Math. 105, 271{282 (1993) [2] Bourbaki, N. : Topologie generale. Chap. III, Groups topologiques (Theorie elementaire). Paris 1960 [3] Heuser, H. : Funktionalanalysis. Stuttgart 1992 [4] Kelley, J. L. : General Topology. Princeton, N.J. 1957
6 56 H. Poppe [5] Naylor, A. W. and Sell G. R. : Linear Operator Theory in Engineering and Science. New York 1982 [6] Poppe, H. : Compactness in General Function Spaces. Berlin 1974 [7] Scheja, G. and Storch, U. : Lehrbuch der Algebra. Teil 2. Stuttgart 1988 [8] Warner, S. : Topological Fields. Amsterdam 1989 received: August 31, 1995 Author: Prof. Dr. H. Poppe Universitat Rostock Fachbereich Mathematik Universitatsplatz Rostock Germany
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