A REMARK ON A PROBLEM OF KLEE

Transcription

1 C O L L O Q U I U M M A T H E M A T I C U M VOL NO. 1 A REMARK ON A PROBLEM OF KLEE BY N. J. K A L T O N (COLUMBIA, MISSOURI) AND N. T. P E C K (URBANA, ILLINOIS) This paper treats a property of topological vector spaces first studied by Klee [6]. X is said to have the Klee property if there are two (ot ecessarily Hausdorff) vector topologies o X, say τ 1 ad τ 2, such that the quasi-orm topology is the supremum of τ 1 ad τ 2 ad such that (X, τ 1 ) has trivial dual while the Hausdorff quotiet of (X, τ 2 ) is early covex, i.e. has a separatig dual. Klee raised the questio of whether every topological vector space has the Klee property. I this paper we will oly cosider the case whe X is a separable quasi- Baach space. I this cotext the problem has recetly bee cosidered i [2] ad [7]. I [7], the problem was cosidered for the special case whe X is a twisted sum of a oe-dimesioal space ad a Baach space, so that there is subspace L of X with dim L = 1 ad X/L locally covex; it was show that X the has the Klee property if the quotiet map is ot strictly sigular. The i [2] a twisted sum X of a oe-dimesioal space ad l 1 was costructed so that the quotiet map is strictly sigular ad X fails to have the Klee property. Thus Klee s questio has a egative aswer. The aim of this paper is to completely characterize the class of separable quasi-baach spaces with the Klee property. Usig this characterizatio we give a much more elemetary couter-example to Klee s questio. Give a quasi-baach space X, the dual of X is deoted by X. We defie the kerel of X to be the liear subspace {x : x (x) = 0 x X }. Now we state our theorem: Theorem. Let X be a separable quasi-baach space, with kerel E. The X fails to have the Klee property if ad oly if E has ifiite codimesio ad the quotiet map π : X X/E is strictly sigular. P r o o f. For the if part, suppose that E has ifiite codimesio, π is strictly sigular ad that X has the Klee property. I this case the closure 1991 Mathematics Subject Classificatio: Primary 46A16. N. Kalto was supported by NSF grat DMS [1]

2 2 N. J. KALTON AND N. T. P E C K of {0} for the topology τ 2 must iclude the kerel E, so that o E the topology τ 1 must coicide with the quasi-orm topology. By a stadard costructio, there is a vector topology τ 3 τ 1 which is pseudo-metrizable ad coicides with the quasi-orm topology o E. Let F be the closure of {0} for this topology. Note that F E = {0} ad that E + F is τ 3 -closed ad hece also closed for the origial topology. This implies that π restricted to F is a isomorphism, ad so if F is of ifiite dimesio, we have a cotradictio. If F is of fiite dimesio, the sice X separates the poits of F we ca write X = X 0 F, ad τ 3 is Hausdorff o X 0. Suppose τ 3 coicides with the origial topology o X 0. The X 0 has trivial dual ad so X 0 E. This implies that E is of fiite codimesio, ad so cotradicts our assumptio. Thus τ 3 is strictly weaker tha the origial topology o X 0. By a theorem of Aoki Rolewicz [4], we ca assume that the quasi-orm is p-subadditive for some 0 < p < 1. Now let deote a F-orm defiig τ 3 o X. There exists δ > 0 so that x δ ad x E imply x p 1/2. We ca also choose 0 < η < 2 1/p so that x i X ad x η together imply that x < δ/2. Also, there exists a sequece (x ) i X 0 so that x 2 ad x = 1. Now by [4] (p. 69, Theorem 4.7) we ca pass to a subsequece, also labelled (x ), which is strogly regular ad M-basic i X, i.e. so that for some M we have max a k M k=1 a kx k for all fiitely ozero sequeces (a k ) k=1. Now pick 0 so large that (M + 1)2 0 < δ/2. Let F 0 be the liear spa of (x ) >0 ; we show that π is a isomorphism o F 0. Ideed, suppose e E ad (a ) >0 is fiitely ozero with e + k> 0 a k x k < η but k> 0 a k x k = 1. The e + k> 0 a k x k δ/2. Further, 1 + max k>0 a k M + 1. Hece e δ ad e p 1/2; this implies e + k> 0 a k x k 1/2, which gives a cotradictio. Hece the map π is a isomorphism o F 0, ad this cotradicts our hypothesis. Now we tur to the coverse. By the theorem of Aoki Rolewicz we ca assume that the quasi-orm is p-subadditive for some 0 < p < 1. Suppose π is a isomorphism o some ifiite-dimesioal closed subspace F. The F has separatig dual ad hece cotais a subspace with a basis. We therefore assume that F has a ormalized basis (f ) =1, ad that K is a costat so large that e E ad f F imply that max( e, f ) K e + f ad so that if (a ) is fiitely ozero the max a K =1 a f. Now let (x ) =1 be a sequece whose liear spa is dese i X, chose i such a way that M p 1 x p = 2 (+4), where each M is a positive iteger. We also require that for each positive iteger m ad each x, mx = αx j for some 0 α 1 ad some positive iteger j. Let N = M M 1, M 0 = 0. Let a = 2 (+4)/p N K 2. Defie V as the absolutely p-covex hull of the set {a f k + x : M k M, 1 < }. We let L be the closed liear spa of the vectors { M k=m 1 +1 f k}.

3 A PROBLEM OF KLEE 3 We ow cosider the set L + V + U X, where U X is the ope uit ball of X. This is a ope absolutely p-covex set ad geerates a p-covex semiquasi-orm o X. We will show that geerates the origial topology o E; more precisely, we will show that if e E ad e L + V + U X, the e 2 1/p K. Ideed, assume e E (L + V + U X ). The there exists y U X so that e y L + V. It follows that there exist fiitely ozero sequeces (b k ) k=1 ad (c ) =1 so that k=1 b k p 1 ad e y = z 1 + z 2, where z 1 = N =1 k=n 1 +1 Let β = N k=n 1 +1 b k. The say. It follows that Thus (b k c )a f k, z 2 = y + z 2 p 1 + max =1 ( N β p x p = A p, =1 z 1 K e z 1 KA. max b k c a K 2 A, M 1 +1 k M k=n 1 +1 so c p K 2p A p a p + b k p (M k M ). Addig, ad usig b k p 1, we obtai Hece, N c p N k=n 1 +1 c p K 2p A p a p Takig pth roots, b k )x. b k p + a p K 2p A p 1 + N a p K 2p A p. + N 1 2N 1 max(n K 2p A p a p, 1). c 2 1/p N 1/p max(1, N 1/p a 1 K 2 A). It ow follows that β 2 1/p N 1 1/p max(1, N 1/p a 1 K 2 A) + N K 2 a 1 A. This implies that β p 3NK p 2p a p A p + 3N p 1. We fially arrive at the iequality A p 1 + A p =1 (3N p K 2p a p + 3N p 1 ) x p Ap,

4 4 N. J. KALTON AND N. T. P E C K which implies A 2 1/p ad hece e K e z 1 KA 2 1/p K, as desired. This shows that L+V +U X itersects E i a bouded set ad iduces the origial topology o E. However, for each, x = 1 N M k=m 1 +1 (a f k + x ) a M f k N k=m 1 +1 is i the covex hull of L + V. Hece, by assumptio o (x ), mx is i the covex hull of L+V as well for all m i N, ad hece (X, ) has trivial dual. Now ote that X with the quasi-orm d(x, E) is early covex. We fially show that the origial topology o X is the supremum of the - topology ad the topology iduced by d(x, E). Suppose d(x, E) 0 ad x 0. The there exist e E so that x e 0 ad so x e 0. Hece e 0, ad hece e 0, which implies x 0. R e m a r k. The mai theorem of [7] is a immediate cosequece of our theorem. Ideed, assume X = R F Y is a twisted sum ad that the quasi-liear map F o the separable ormed space Y splits o a ifiitedimesioal subspace. The it is bouded o a further ifiite-dimesioal subspace, so the quotiet map π : X X/E = X/R is ot strictly sigular. R e m a r k. Oe special case, which is sometimes applicable, is that X has the Klee property if it has a ifiite-dimesioal locally covex subspace with the Hah Baach Extesio Property (cf. [4]). Ideed, i these circumstaces, there is a locally covex subspace Z with dim Z =, so the Baach evelope semiorm is equivalet to the origial quasi-orm o Z; it the follows rapidly that the quotiet map π : X X/E is a isomorphism o Z. For a particular case of this, let (A ) be a sequece of pairwise disjoit measurable subsets of (0, 1), of positive measure. Let (f ) be a sequece of measurable fuctios, with f supported o A ad each f havig the distributio of t 1/t, for small t. Let F be the closed liear spa of (f ) i weak L 1. The F has the Hah Baach Extesio Property i weak L 1 ad so if X is ay separable subspace of weak L 1 cotaiig F the X has the Klee property. Example. Fially, we costruct a elemetary couter-example to Klee s problem, usig much less techical argumets tha [2]. We use the twisted sum of Hilbert spaces, Z 2, itroduced i [3] (see alterative treatmets i [1] ad [5]). To defie this it will be coveiet to cosider the space c 00 of all fiitely ozero sequeces as a dese subspace of l 2 ad cosider the map Ω : c 00 l 2 give by Ω(ξ)(k) = ξ(k) log( ξ 2 / ξ(k) ),

5 A PROBLEM OF KLEE 5 where as usual the right-had side is iterpreted as zero if ξ(k) = 0. The Ω(αξ) = αω(ξ) for α R ad Ω(ξ + η) Ω(ξ) Ω(η) 2 C( ξ 2 + η 2 ) for a suitable absolute costat C. Now Z 2 = l 2 Ω l 2 is the completio of c 00 l 2 uder the quasi-orm (ξ, η) = ξ Ω(η) 2 + η 2. Now (cf. [3]) the map (ξ, η) η exteds to a quotiet map from Z 2 oto l 2 which is strictly sigular. More precisely, if F is ay ifiite-dimesioal subspace of c 00 the the completio of l 2 Ω F cotais a isometric copy of Z 2 (this is essetially Theorem 6.5 of [3], or see [1]). I particular, this subspace is ever of cotype 2. Now to costruct our example, embed l 2 ito L p, where p < 1. The L p Ω l 2 = X has its kerel E isomorphic to L p ad X/E l 2. If the quotiet map is ot strictly sigular the there is a ifiite-dimesioal subspace F of c 00 such that the completio of L p Ω F is liearly isomorphic to L p l 2 ad hece has cotype 2. The l 2 Ω F is also cotype 2, ad this is impossible as we have see. It follows from our mai theorem that the space we have costructed fails the Klee property. REFERENCES [1] N. J. Kalto, The space Z 2 viewed as a symplectic Baach space, i: Proc. Research Workshop i Baach Space Theory, Uiv. of Iowa, Iowa City, 1981, [2], The basic sequece problem, Studia Math. 116 (1995), [3] N. J. Kalto ad N. T. Peck, Twisted sums of sequece spaces ad the three-space problem, Tras. Amer. Math. Soc. 255 (1979), [4] N. J. Kalto, N. T. Peck ad J. W. Roberts, A F-space Sampler, Lodo Math. Soc. Lecture Note Ser. 89, Cambridge Uiv. Press, Cambridge, [5] N. J. Kalto ad R. C. Swaso, A symplectic Baach space with o Lagragia subspace, Tras. Amer. Math. Soc. 273 (1982), [6] V. L. Klee, Exotic topologies for liear spaces, i: Proc. Sympos. o Geeral Topology ad its Relatios to Moder Aalysis ad Algebra, Academic Press, 1962, [7] N. T. Peck, Twisted sums ad a problem of Klee, Israel J. Math. 81 (1993), Departmet of Mathematics Departmet of Mathematics Uiversity of Missouri Uiversity of Illiois Columbia, Missouri Urbaa, Illiois U.S.A. U.S.A. mathjk@mizzou1.bitet peck@symcom.math.uiuc.edu Received 6 February 1995