Networks for the weak topology of Banach and Fréchet spaces
aa r X i v : . [ m a t h . F A ] D ec NETWORKS FOR THE WEAK TOPOLOGY OF BANACH ANDFR ´ECHET SPACES
S. GABRIYELYAN, J. KA¸ KOL, W. KUBI´S, AND W. MARCISZEWSKI
Dedicated to the Memory of Professor Manuel Valdivia
Abstract.
We start the systematic study of Fr´echet spaces which are ℵ -spaces inthe weak topology. A topological space X is an ℵ -space or an ℵ -space if X has acountable k -network or a σ -locally finite k -network, respectively. We are motivatedby the following result of Corson (1966): If the space C c ( X ) of continuous real-valued functions on a Tychonoff space X endowed with the compact-open topologyis a Banach space, then C c ( X ) endowed with the weak topology is an ℵ -space if andonly if X is countable. We extend Corson’s result as follows: If the space E := C c ( X )is a Fr´echet lcs, then E endowed with its weak topology σ ( E, E ′ ) is an ℵ -space ifand only if ( E, σ ( E, E ′ )) is an ℵ -space if and only if X is countable. We obtain anecessary and some sufficient conditions on a Fr´echet lcs to be an ℵ -space in the weaktopology. We prove that a reflexive Fr´echet lcs E in the weak topology σ ( E, E ′ ) is an ℵ -space if and only if ( E, σ ( E, E ′ )) is an ℵ -space if and only if E is separable. Weshow however that the nonseparable Banach space ℓ ( R ) with the weak topology isan ℵ -space. Introduction
Topological properties of a locally convex space (lcs for short) E in the weak topology σ ( E, E ′ ) are of the importance and have been intensively studied from many years (see[18, 25]). Corson (1961) started a systematic study of certain topological propertiesof the weak topology of Banach spaces. This line of research provided more generalclasses such as reflexive Banach spaces, weakly compactly generated Banach spaces( W CG
Banach spaces) and the class of weakly K -analytic and weakly K -countablydetermined Banach spaces. We refer the reader to [11] and [18] for many referencesand facts.Although ( E, σ ( E, E ′ )) is never a metrizable space for a separable infinite dimen-sional normed E , every σ ( E, E ′ )-compact set is σ ( E, E ′ )-metrizable (see [18, Corollary4.6] or [11, Proposition 3.29]). Moreover, for many natural and important classes of Mathematics Subject Classification.
Primary 46A03, 54H11; Secondary 22A05, 54C35.
Key words and phrases.
Fr´echet space, weakly ℵ locally convex space, ℵ -space, ℵ -space, space ofcontinuous functions.The first named author was partially supported by Israel Science Foundation grant 1/12.The second named author was supported by Generalitat Valenciana, Conselleria d’Educaci´o, Cul-tura i Esport, Spain, Grant PROMETEO/2013/058. Part of this work has been done when the secondnamed author visited Department of Mathematics Czech Academy of Science as a Visiting Professor,April 2014.The third named author was supported by GA ˇCR grant No. P201 14 07880S.The fourth named author was supported by National Science Center research grant DEC-2012/07/B/ST1/03363. separable metrizable lcs E , the space ( E, σ ( E, E ′ )) is a generalized metric space ofsome type (see [2, 13, 20]). Such types of topological spaces are defined by differenttypes of networks. The concept of network is one of a well recognized good tool, com-ing from the pure set-topology, which turned out to be of great importance to studysuccessfully renorming theory in Banach spaces, see the survey paper [6]; especially [6,Theorem 13] for σ ( E, F )-slicely networks.Following Michael [20], a family N of subsets of a topological space X is called a k -network in X if whenever K ⊂ U with K compact and U open in X , then K ⊂ S F ⊂ U for some finite F ⊂ N . A topological space X is said to be an ℵ -space if X is regular and has a countable k -network [20]. It is known that a regular space isan ℵ -space if and only if it is a continuous image of a separable metric space undera compact-covering mapping ([20]). Every ℵ -space is separable and Lindel¨of. It isknown that every Banach space E whose strong dual E ′ is separable is a weakly ℵ -space , i.e. E with the weak topology σ ( E, E ′ ) is an ℵ -space, see [20] and [2] (or [13]for more general facts for Fr´echet lcs , i.e. metrizable and complete lcs).O’Meara [22] generalized the concept of ℵ -spaces as follows: A topological space X is called an ℵ -space if it is regular and has a σ -locally finite k -network. Any metrizablespace is an ℵ -space and all compact sets in ℵ -spaces are metrizable. For further results,see [15]. The study of those locally convex spaces E which are weakly ℵ -spaces (i.e.under the weak topology E has a σ -locally finite k -network) is begun here for theimportant particular case of Fr´echet lcs.If X is a Tychonoff space, C c ( X ) (resp. C p ( X )) denotes the space C ( X ) of allcontinuous functions on X endowed with the compact-open (resp. pointwise) topology.It is well known that C c ( X ) is metrizable if and only if X is hemicompact , i.e. X admitsa fundamental sequence of compact sets, see [1]. Moreover, C c ( X ) is complete if andonly if X is a k R -space, see [25]. Note that C p ( X ) is an ℵ -space if and only if X iscountable [27]. Corson [20] proved the following interesting result (*): If K is compact,the Banach space C ( K ) is a weakly ℵ -space if and only if K is countable. Our mainresult extends Corson’s theorem. Theorem 1.1.
A Fr´echet lcs C c ( X ) is a weakly ℵ -space if and only if C c ( X ) is aweakly ℵ -space if and only if X is countable. If X is a countable and locally compact space, Theorem 1.1 guarantees that C c ( X )is even a weakly ℵ -space. We note the following question: Is C c ( X ) a weakly ℵ -spacefor any countable Tychonoff space X ? Having in mind that the weak topology of C c ( X ) lies between the compact open topology and the pointwise one, the question isespecially interesting for the case X is an ℵ -space. Recall that for such X the spaces C c ( X ) and C p ( X ) are ℵ -spaces by [20], and C p ( X ) is even separable and metrizable.In Section 7 we prove that C c ( X ) is a weakly ℵ -space for any countable ℵ -space X .Although ℵ -spaces and ℵ -spaces are essentially different, in the class of Lindel¨ofspaces they coincide, see Proposition 5.9 below. Therefore, it is interesting also todescribe possible large classes of Fr´echet (or Banach) spaces for which the both conceptscoincide for the weak topology. We observe that any W CG
Banach space is a weakly ℵ -space if and only if it is a weakly ℵ -space (Corollary 5.11). We show that a Banachspace E not containing a copy of ℓ is a weakly ℵ -space if and only if it is a weakly ℵ -space if and only if the strong dual E ′ of E is separable (Corollary 5.6). This extends ETWORKS FOR THE WEAK TOPOLOGY 0F BANACH AND FR´ECHET SPACES 3 a corresponding result for the case ℵ -spaces (see [2, §
12] and [13]). Consequently, forany 1 < p < ∞ and an uncountable set Γ the (reflexive) Banach space ℓ p (Γ) is not aweakly ℵ -space. We show even more: A reflexive Fr´echet lcs E is a weakly ℵ -space ifand only if E is a weakly ℵ -space if and only if E is separable (Corollary 6.6). Theseresults motivate the following natural question: Does there exist a nonseparable Banachspace E which is an ℵ -space in the weak topology of E ? We answer this question inthe affirmative by proving the following theorem.
Theorem 1.2.
The Banach space ℓ (Γ) is an ℵ -space in the weak topology if and onlyif the cardinality of Γ does not exceed the continuum. So, the nonseparable Banach space ℓ ( R ) endowed with the weak topology is an ℵ -space but is not an ℵ -space. Moreover, the space ℓ ( R ) in the weak topology is notnormal, see Proposition 4.7.2. Some definitions and known facts
Recall (see [14]) that a family N of subsets of a topological space X is a cs ∗ -networkat a point x ∈ X if for each sequence ( x n ) n ∈ N in X converging to x and for eachneighborhood O x of x there is a set N ∈ N such that x ∈ N ⊂ O x and the set { n ∈ N : x n ∈ N } is infinite (where N = { , , . . . } ); N is a cs ∗ -network in X if N isa cs ∗ -network at each point x ∈ X . The smallest size |N | of a cs ∗ -network N at x iscalled the cs ∗ -character of X at the point x and is denoted by cs ∗ χ ( X, x ). The cardinal cs ∗ χ ( X ) = sup { cs ∗ χ ( X, x ) : x ∈ X } is called the cs ∗ -character of X . Recall also (see[20]) that a point x in a topological space X is called an r -point if there is a sequence { U n } n ∈ N of neighborhoods of x such that if x n ∈ U n , then { x n } n ∈ N has compact closure;call X an r -space if all of its points are r -points. The first countable spaces and thelocally compact spaces are r -spaces. Theorem 2.1 ([23]) . A topological space X is metrizable if and only if it is an ℵ -spaceand an r -space. A topological space X has the property ( α ) at a point x ∈ X if for any { x m,n :( m, n ) ∈ N × N } ⊂ X with lim n x m,n = x ∈ X , m ∈ N , there exists a sequence( m k ) k of distinct natural numbers and a sequence ( n k ) k of natural numbers such thatlim k x m k ,n k = x ; X has the property ( α ) or is an ( α ) -space if it has the property ( α )at each point x ∈ X . Nyikos proved in [24, Theorem 4] that any Fr´echet-Urysohntopological group satisfies ( α ). However there are Fr´echet-Urysohn topological spaceswhich do not have ( α ) (see for instance Example 2.4).Theorem 2.1 combined with additional facts from [3] yields also Theorem 2.2. An ℵ -space X is metrizable if and only if X is a Fr´echet-Urysohn ( α ) -space.Proof. Clearly, if X is metrizable then it is a Fr´echet-Urysohn ( α )-space. Conversely,let X be a Fr´echet-Urysohn ( α )-space. Being an ℵ -space, X has countable cs ∗ -character (this might be also noticed from the proof of [27, Corollary 2.18]). Indeed,it immediately follows from the definitions of k - and cs ∗ -networks that any closed k -network is a cs ∗ -network. So it is enough to show that any space X with a σ -locallyfinite closed cs ∗ -network D = S n ∈ N D n has countable cs ∗ -character. Fix x ∈ X . For S. GABRIYELYAN, J. KA¸ KOL, W. KUBI´S, AND W. MARCISZEWSKI every n ∈ N set T n ( x ) := { D ∈ D n : x ∈ D } . Since D n is locally finite, the family T n ( x ) is finite. So the family T ( x ) := S n ∈ N T n ( x ) is countable. We show that T ( x ) isa countable cs ∗ -network at x . Let x n → x and U be a neighborhood of x . Since D is a cs ∗ -network, there is k ∈ N and D ∈ D k such that x ∈ D ⊂ U and D containsinfinitely many elements of { x n } n ∈ N . As D is closed, it contains x , so D ∈ T k ( x ). Now[3, Proposition 6, Lemma 7] imply that X is first countable, hence an r -space. Finally, X is metrizable by Theorem 2.1. (cid:3) Since every Fr´echet-Urysohn topological group satisfies property ( α ) by [24, Theo-rem 4], we obtain Corollary 2.3.
A topological group G is metrizable if and only if G is a Fr´echet-Urysohn ℵ -space. Example 2.4.
Let V ( ℵ ) be the Fr´echet-Urysohn fan which is obtained by the iden-tifying the limit points of the compact spaces S n , where S n = (cid:8) , k (cid:9) k ∈ N ⊂ R . It iswell-known that any compact subset of V ( ℵ ) is contained in a finite union of sequences S n . Hence the natural map from the topological direct sum L k ∈ N S k onto V ( ℵ ) iscompact-covering. So V ( ℵ ) is an ℵ -space [20]. Clearly, V ( ℵ ) is not metrizable andnot an ( α )-space (and it is not an r -space by Theorem 2.1).3. Some necessary conditions for being an ℵ -space Recall that a topological space X is called a σ -space if X is regular and has a σ -discrete (equivalently, σ -locally finite) network. If X is regular and has a countablenetwork, X is called a cosmic space . Clearly ℵ -spaces and cosmic spaces are σ -spaces.It is well known (see [15]) that any closed subset H of a σ -space X is a G δ -set. Indeed,if S n ∈ N D n is a σ -discrete closed network for X , then the sets A n := S { D ∈ D n : D ∩ H = ∅} are closed in X by [9, 1.1.11]. As H = T n ∈ N ( X \ A n ), H is a G δ -set.Consequently, any σ -space has countable pseudocharacter; we denote ψ ( X ) = ℵ .Clearly, every separable Banach space with the Schur property is a weakly ℵ -space.In Section 4 we show that ℓ ( R ) is a weakly ℵ -space.It is well known that the dual space of ℓ p (Γ) is ℓ q (Γ), where 1 /p + 1 /q = 1. So, thesupport of continuous functionals over ℓ p (Γ) must be countable. We use this fact toprove the following Example 3.1.
Let Γ be an infinite set and E := ℓ p (Γ) with < p < ∞ . Then ψ ( E w ) ≥ | Γ | , where E w := ( E, σ ( E, E ′ )) . Hence ℓ p (Γ) are not weakly σ -spaces forevery uncountable Γ .Proof. If Γ is countable the assertion is clear. Suppose that Γ is uncountable. Let U = { U i } i ∈ I be a family of weakly open neighborhoods of 0 such that T i ∈ I U i = { } and | I | = ψ ( E w ). We may assume that each U i has the following standard form U i = { x ∈ E : | χ i,k ( x ) | < δ i , where χ i,k ∈ E ′ for 1 ≤ k ≤ m i } . Suppose, for a contradiction, that | I | < | Γ | . Denote by J the set of all indices j ∈ I such that the j -coordinate is nonzero for some χ i,k . So | J | = | N | × | N | × | I | < | Γ | .Hence we can find an index γ ∈ Γ \ J . Set x = ( r γ ) γ ∈ Γ , where r γ = 1 if γ = γ , and r γ = 0 otherwise. Clearly, x ∈ E and χ i,k ( x ) = 0 for all i ∈ I and every 1 ≤ k ≤ m i ,that contradicts the choice of the family U . Thus | I | ≥ | Γ | . (cid:3) ETWORKS FOR THE WEAK TOPOLOGY 0F BANACH AND FR´ECHET SPACES 5
We provide some necessary condition for any lcs to be a weakly ℵ -space. First weprove the following useful observation. Lemma 3.2.
Let E be a non-trivial lcs. Then E w := ( E, σ ( E, E ′ )) has countablepseudocharacter if and only if E w admits a weaker separable metrizable lcs topology.In particular, | E | = c provided E w has countable pseudocharacter.Proof. Assume that E w has countable pseudocharacter. Let T n ∈ N U n = { } , where theopen sets U n have the following standard form U n = { x ∈ E : | χ i,n ( x ) | < δ n , where χ i,n ∈ E ′ for 1 ≤ i ≤ k n } . Let { χ n } n ∈ N be an enumeration of the family { χ i,n : 1 ≤ i ≤ k n , n ∈ N } . Then T n ∈ N ker( χ n ) = { } . This implies that the following map p : E w → Y n ∈ N E/ ker( χ n ) = R N , p ( x ) = ( χ n ( x )) n ∈ N , is continuous and injective, and hence | E | = c as E is non-trivial. Now the topologyinduced on E w from R N is as desired. The converse assertion is trivial. (cid:3) Next fact is well known but hard to locate.
Lemma 3.3.
A lcs E admits a metrizable and separable locally convex topology τ weaker than σ ( E, E ′ ) if and only if ( E ′ , σ ( E ′ , E )) is separable. Lemmas 3.2 and 3.3 imply the following necessary conditions on lcs E to be weakly ℵ -space which partially converse Proposition 5.2(i). Proposition 3.4. If E is a non-trivial lcs which is a weakly σ -space, then (i) ( E, σ ( E, E ′ )) admits a weaker separable metrizable lcs topology; (ii) ψ ( E, σ ( E, E ′ )) = ℵ and | E | = c ; (iii) ( E ′ , σ ( E ′ , E )) is separable. Note that the space ℓ ∞ satisfies above conditions (i)–(iii), although it is not a weakly ℵ -space (see Corollary 6.7 below).4. ℓ ( R ) is an ℵ -space in the weak topology In this section we prove Theorem 1.2 which states that the Banach space ℓ (Γ) isan ℵ -space in the weak topology if and only if the cardinality of Γ does not exceedthe continuum. In particular, the space ℓ ( R ) is a weakly ℵ -space. Clearly, the “onlyif” part of the theorem follows from Proposition 3.4 because the space ℓ (Γ) with theweak topology does not have countable pseudocharacter whenever | Γ | > ℵ . Theremaining part of this section is devoted to the proof of the “if” part. It is clear that ifthe cardinality of the set Γ is less than or equal to the cardinality of the set Γ then ℓ (Γ ) embeds into ℓ (Γ ), therefore it is enough to consider the case when Γ has thecardinality continuum.We shall work with the space ℓ (2 ω ), where 2 ω denotes the Cantor set, treated justas an index set of cardinality continuum (recall that the space ℓ ( S ) does not dependon any extra structure of the set S ).We shall use some ideas from [19] (especially from the proof of Lemma 2.3.1 in [19]).Given a Banach space E , we shall denote by B E and S E the closed unit ball and theunit sphere of E , respectively. S. GABRIYELYAN, J. KA¸ KOL, W. KUBI´S, AND W. MARCISZEWSKI
Lemma 4.1.
The unit sphere S ℓ ∞ (2 ω ) is weak ∗ separable.Proof. Let P be the family of all (necessarily finite) partitions of the Cantor set intofinitely many open sets. As 2 ω is zero-dimensional, for every finite set F ⊂ ω there is P ∈ P such that P = { U x : x ∈ F } and x ∈ U x for every x ∈ F . Obviously, the family P is countable, because the Cantor set has only countably many sets that are open andclosed simultaneously. Define D = (X U ∈ P q U χ U ∈ S ℓ ∞ (2 ω ) : P ∈ P , { q U : U ∈ P } ⊂ Q ) , where χ A denotes the characteristic function of a set A . Obviously, D is countable.We claim that it is weak ∗ dense in S ℓ ∞ (2 ω ) .In fact, given x , . . . , x k ∈ ℓ (2 ω ) and ε >
0, a basic weak ∗ neighborhood of y ∈ S ℓ ∞ (2 ω ) is of the form V = { v ∈ S ℓ ∞ (2 ω ) : | v ( x i ) − y ( x i ) | < ε for i = 1 , , . . . , k } . Fix δ > F ⊂ ω be a finite set such that(4.1) k x i k − X t ∈ F | x i ( t ) | = X t F | x i ( t ) | < δ for every i = 1 , , . . . , k . Take a partition P ∈ P such that U ∩ F is either empty or asingleton, whenever U ∈ P , and there is U ∈ P such that U ∩ F = ∅ . For every t ∈ F and each U ∈ P containing t ∈ U take q U ∈ [ − , ∩ Q such that | q U − y ( t ) | < δ , andset q U = 1 for every U ∈ P such that U ∩ F = ∅ . Set w = P U ∈ P q U χ U . Then w ∈ D .We show that w ∈ y + V for δ small enough. Indeed, for every i = 1 , , . . . , k , theinequality (4.1) and the construction of w imply | w ( x i ) − y ( x i ) | ≤ X t ∈ F | w ( t ) x i ( t ) − y ( t ) x i ( t ) | + X t F | w ( t ) x i ( t ) − y ( t ) x i ( t ) | < δ · X t ∈ F | x i ( t ) | + X t F | x i ( t ) | < δ ( k x i k + 2) . Now it is clear that if δ is small enough then w ∈ y + V . Thus S ℓ ∞ (2 ω ) is weak ∗ separable. (cid:3) Lemma 4.2.
Let E be a Banach space such that ( S E ′ , weak ∗ ) is separable. Then forevery r > there exists a countable family F of weakly closed subsets of E containedin E \ r B E = { x ∈ E : k x k > r } and such that E \ r B E = [ F ∈F int w ( F ) , where int w denotes the interior with respect to the weak topology.Proof. Let D be a countable weak ∗ dense subset of S E ′ . Given ϕ ∈ D , n ∈ N , define F ϕ,n = { x ∈ E : ϕ ( x ) ≥ r + 1 /n } . Then F = { F ϕ,n : ϕ ∈ D, n ∈ N } is the required family. (cid:3) ETWORKS FOR THE WEAK TOPOLOGY 0F BANACH AND FR´ECHET SPACES 7
Lemma 4.3.
Let < ε < r and let M ( r, ε ) = { x ∈ ℓ (2 ω ) : r − ε < k x k ≤ r } . Then for every x ∈ M ( r, ε ) there exists a weakly (in fact, pointwise) open set V ⊂ ℓ (2 ω ) such that x ∈ V and diam( V ∩ M ( r, ε )) ≤ ε .Proof. Given A ⊂ ω , denote by p A the canonical projection from ℓ (2 ω ) onto ℓ ( A ),that is, p A ( v ) = v ↾ A . Fix x ∈ M ( r, ε ). There exists a finite set F ⊂ ω suchthat k p F ( x ) k > r − ε . Choose an open set U ⊂ ℓ ( F ) such that k u k > r − ε and k u − p F ( x ) k < ε for every u ∈ U . Let V = p − F ( U ). We claim that V is as required.Obviously, x ∈ V and V is pointwise (in particular, weakly) open. Fix y , y ∈ V ∩ M ( r, ε ). Let A = 2 ω \ F .Note that the ℓ -norm has the property that k v k = k p F ( v ) k + k p A ( v ) k for every v ∈ ℓ (2 ω ) . In particular, k p A ( y i ) k ≤ ε , because k p F ( y i ) k > r − ε and k y i k ≤ r for i = 1 ,
2. Using these facts we get k y − y k = k p F ( y ) − p F ( y ) k + k p A ( y ) − p A ( y ) k≤ k p F ( y ) − p F ( x ) k + k p F ( x ) − p F ( y ) k + k p A ( y ) k + k p A ( y ) k≤ ε. It follows that diam( V ∩ M ( r, ε )) ≤ ε . (cid:3) The next statement is rather standard; it has been used implicitly, e.g., in [19].
Lemma 4.4.
Let X be a metric space. Then there exists an open base B in X suchthat B = S n ∈ N B n and each B n is uniformly discrete, that is, for every n there is ε n > such that the distance of any two distinct members of B n is > ε n .Proof. A theorem of Stone says that every open cover of a metric space X admits a σ -discrete open refinement. The proof (see, e.g., [9, Proof of Thm. 4.4.1]) actuallyshows that every open cover of X has an open refinement of the form U = S n ∈ N U n ,where each U n is uniformly discrete. Now let B = S n ∈ N W n be such that W n is an openrefinement of a cover by balls of radius 1 /n and W n is a countable union of uniformlydiscrete families. Then B is easily seen to be an open base. (cid:3) Remark 4.5.
The proof of Theorem 1.2 uses the well known fact stating that the space ℓ (Γ) has the Schur property (that is any convergent sequence in the weak topology isalso a convergent sequence in the norm topology) for every set Γ, see [11]. This impliesthat any weakly compact set of ℓ (Γ) is also norm compact. Proof of Theorem 1.2.
Let B = S n ∈ N B n be a base of open sets for the norm topologyon ℓ (2 ω ) such that the distance between every two distinct members of B n is > /k n for every n ∈ N (here we have used Lemma 4.4).Given r >
0, define U r = ℓ (2 ω ) \ r B ℓ (2 ω ) . Let F r be a countable family of weakly closed subsets of U r such that S F ∈F r int w F = U r (Lemma 4.2). Let F r = { F mr } m ∈ N . S. GABRIYELYAN, J. KA¸ KOL, W. KUBI´S, AND W. MARCISZEWSKI
Given n, m, i ∈ N , define L ( i, n ) := M (cid:18) i + 210 k n , k n (cid:19) = (cid:26) x ∈ ℓ (2 ω ) : i k n < k x k ≤ i + 210 k n (cid:27) and C ( n, m, i ) = { B ∩ F mi/ k n ∩ L ( i, n ) : B ∈ B n } . Claim 4.6.
For every n, m, i ∈ N , the family C ( n, m, i ) is discrete in the weak topology.Proof. Note that the union of C ( n, m, i ) is contained in the weakly closed set F mi/ k n ∩ (( i + 2) / k n ) B ℓ (2 ω ) , therefore it is enough to show that every point of this set has a weak neighborhoodmeeting at most one set from C ( n, m, i ).Fix x ∈ F mi/ k n ∩ (( i + 2) / k n ) B ℓ (2 ω ) ⊂ L ( i, n ). By Lemma 4.3, there exists aweakly open set V such that x ∈ V anddiam (cid:0) V ∩ L ( i, n ) (cid:1) ≤ / k n < /k n . The set V can intersect at most one B ∈ B n , as B n is 1 /k n -discrete. (cid:3) Let O ( r ) = { x ∈ ℓ (2 ω ) : k x k < r } and define D ( n ) = { B ∩ O (1 / k n ) : B ∈ B n } . Note that actually D ( n ) contains at most one nonempty set (because 2 / k n < /k n ),therefore it is certainly discrete in the weak topology.Define A = [ {C ( n, m, i ) : n, m, i ∈ N } ∪ {D ( n ) : n ∈ N } . Then the family A is σ -discrete with respect to the weak topology. It remains toshow that A is a k -network in ℓ (2 ω ) with the weak topology.Fix a weakly compact set K contained in a weakly open set U ⊂ ℓ (2 ω ). It followsfrom Remark 4.5 that K is also compact in the norm topology. Choose a finite E ⊂ B such that K ⊂ S E ⊂ U . Choose n such that E ⊂ S n n =1 B n .For i, n ∈ N , let N ( i, n ) := int (cid:0) L ( i, n ) (cid:1) . Note that, for each n ∈ N , the space ℓ (2 ω )is covered by O (1 / k n ) and the sets N ( i, n ), i ∈ N . Given B ∈ B n , by Lemma 4.2 wehave that B = [ (cid:8) B ∩ int w ( F mi/ k n ) ∩ N ( i, n ) : m, i ∈ N (cid:9) ∪ ( B ∩ O (1 / k n )) . Therefore the family [ n ≤ n (cid:0)(cid:8) B ∩ int w ( F mi/ k n ) ∩ N ( i, n ) : B ∈ B n ∩ E , m, i ∈ N (cid:9) ∪ { B ∩ O (1 / k n ) : B ∈ B n ∩ E } )covers K and consists of norm open sets, so it has a finite subfamily covering K . Hence,for any n ≤ n , we can find i ( n ) and m ( n ) such that the finite subfamily F := [ n ≤ n (cid:0)(cid:8) B ∩ F mi/ k n ∩ L ( i, n ) : B ∈ B n ∩ E , m ≤ m ( n ) , i ≤ i ( n ) (cid:9) ∪{ B ∩ O (1 / k n ) : B ∈ B n ∩ E } (cid:1) ETWORKS FOR THE WEAK TOPOLOGY 0F BANACH AND FR´ECHET SPACES 9 of A satisfies K ⊂ S F ⊂ S E ⊂ U . Thus the σ -locally finite family A is also a k -network for ℓ (2 ω ) in the weak topology. (cid:3) Proposition 4.7.
The space ℓ ( R ) endowed with the weak topology is not normal.Proof. Suppose for a contradiction that ℓ ( R ) with the weak topology is a normal space.Then the square ℓ ( R ) of ℓ ( R ) with the weak topology ω is also normal (note that ℓ ( R ) and ℓ ( R ) endowed with the weak topologies are homeomorphic). Now Corson’slemma [7, Lemma 7] applies to derive that every ω -discrete set in ℓ ( R ) is countable,which clearly leads to a contradiction. (cid:3) Remark 4.8.
Recall that Foged in [12] constructed already a non-normal space whichis an ℵ -space. Our example of such a space seems to be however very natural and usesa well known Banach space ℓ ( R ). Also O’Meara [21] gave an example (unpublished) ofan ℵ -space which is not paracompact. The authors thank to Professor Gary Gruenhagefor providing references included in the above remark.Notice also that the proof of Theorem 1.2 essentially uses the fact that ℓ ( R ) hasthe Schur property (see Remark 4.5). Therefore it is natural to ask: Question 4.9.
Let E be a Banach space with the Schur property and satisfy (i)-(iii)of Proposition 3.4. Is E an ℵ -space in the weak topology? Taking into account Remark 4.5, every separable Banach space with the Schurproperty in the weak topology is an ℵ -space.5. Interplay between weakly ℵ and weakly ℵ -Fr´echet spaces Recall that a lcs E is called trans-separable if for each neighborhood of zero U in E there exists a countable subset N of E such that E = N + U . Clearly for metrizablelcs trans-separability and separability are equivalent concepts. Lemma 5.1 ([18, Cor. 6.8]) . The strong dual of a lcs E is trans-separable if and onlyif every bounded set in E is metrizable in the weak topology of E . Recall that a Fr´echet lcs E satisfies the density condition if every bounded set in E ′ (with the strong topology) is metrizable (cf. [18, Prop. 6.16]). The class of suchspaces includes Fr´echet-Montel locally convex spaces and quasinormable Fr´echet locallyconvex spaces. The latter class contains all Banach spaces, as well as every ( F S )-space(see [5]). In [13] we proved the following
Proposition 5.2 ([13]) . Let E be a Fr´echet lcs and E ′ be its strong dual. Then (i) If E ′ is separable, then E is a weakly ℵ -space. (ii) If E is a weakly ℵ -space not containing a copy of ℓ , then E ′ is trans-separable. (iii) If E is a weakly ℵ -space, then E ′ is trans-separable if and only if every boundedset in E is Fr´echet-Urysohn in the weak topology of E . (iv) If E satisfies the density condition and does not contain a copy of ℓ , then E isa weakly ℵ -space if and only if E ′ is separable. (v) If E does not contain a copy of ℓ , then every bounded set in E is Fr´echet-Urysohn in σ ( E, E ′ ) . Corollary 5.3.
A reflexive Fr´echet lcs E is a weakly ℵ -space if and only if E isseparable (if and only if E is a weakly ℵ -space). Proof. As E is reflexive, ( E ′ , σ ( E ′ , E )) is separable if and only if ( E ′ , β ( E ′ , E )) is sep-arable. Assume that E is a weakly ℵ -space. Then ( E ′ , σ ( E ′ , E )) is separable byProposition 3.4, so ( E ′ , β ( E ′ , E )) is separable. By Proposition 5.2(i) the space E isa weakly ℵ -space. In particular, E is separable. Conversely, if E is separable then( E ′ , σ ( E ′ , E )) is separable and Proposition 5.2 (i) applies. (cid:3) Since every nuclear Fr´echet space is a separable reflexive space, see [5], we have
Corollary 5.4.
Every nuclear Fr´echet space is a weakly ℵ -space. We apply Theorem 2.2 to extend parts (ii) and (iii) of Proposition 5.2.
Theorem 5.5.
Let E be a lcs which is a weakly ℵ -space. Then the strong dual E ′ of E is trans-separable if and only if every bounded set in E is Fr´echet-Urysohn in theweak topology of E . If in addition E is a Fr´echet lcs not containing a copy of ℓ , then E ′ is trans-separable.Proof. If E ′ is trans-separable, then every bounded set in E is metrizable in σ ( E, E ′ )by Lemma 5.1. Conversely, if every bounded set in E is Fr´echet-Urysohn in σ ( E, E ′ ),apply [13, Lemma 3.2] to see that every bounded set B in E is a Fr´echet-Urysohn ( α )-space in σ ( E, E ′ ). As a subspace of the ℵ -space ( E, σ ( E, E ′ )), B is also an ℵ -space. ByTheorem 2.2, B is metrizable. Finally, Lemma 5.1 applies to get the trans-separabilityof E ′ . The last assertion follows from the first one and Proposition 5.2(v). (cid:3) As the strong dual of a Banach space is normed, this theorem combined with Propo-sition 5.2 yield the following
Corollary 5.6.
Let E be a Banach space not containing a copy of ℓ . Then E is aweakly ℵ -space if and only if E is a weakly ℵ -space if and only its strong dual E ′ isseparable. Any reflexive Fr´echet lcs E does not contain a copy of ℓ , but E may not satisfy thedensity condition [4]. The following result generalizes (iv) of Proposition 5.2. Theorem 5.7.
Let E be a Fr´echet lcs not containing a copy of ℓ and satisfying thedensity condition. Then E is a weakly ℵ -space if and only if E is a weakly ℵ -space ifand only if the strong dual of E ′ of E is separable.Proof. Clearly, the strong dual E ′ is a ( DF )-space, see [25, Theorem 8.3.9], with afundamental sequence ( Q n ) n of absolutely convex bounded subsets of E ′ . Since E satisfies the density condition, every bounded set Q n is metrizable by [5, Corollary 3].Assume now that E is a weakly ℵ -space. By Theorem 5.5 the strong dual E ′ is trans-separable. So the trans-separable lcs E ′ is covered by a sequence of metrizable boundedabsolutely convex sets ( Q n ) n . Now Corollary 4.12 of [13] implies that E ′ is separable.As E ′ is separable, then E is a weakly ℵ -space by Proposition 5.2(i). Finally, if E isa weakly ℵ -space it is also a weakly ℵ -space. (cid:3) Since every Fr´echet lcs C c ( X ) satisfies the density condition (see [26] or [5]), weapply Theorem 5.7 to get Corollary 5.8.
Let E := C c ( X ) be a Fr´echet lcs not containing a copy of ℓ . Then E is a weakly ℵ -space if and only if E is a weakly ℵ -space if and only if the strong dual E ′ of E is separable. ETWORKS FOR THE WEAK TOPOLOGY 0F BANACH AND FR´ECHET SPACES 11
We need the following useful fact, see also [22] for (ii).
Proposition 5.9.
Let X be a topological space. (i) X is a cosmic space if and only if X is a Lindel¨of σ -space. (ii) X is an ℵ -space if and only if X is a Lindel¨of ℵ -space.Proof. Assume that X is a Lindel¨of σ -space (respectively, an ℵ -space) with a σ -locallyfinite network (respectively, k -network) D = S n ∈ N D n . It is enough to prove that every D n is countable. For every x ∈ X choose an open neighborhood U x of x such that U x intersects with a finite subfamily T ( x ) of D n . Since X is a Lindel¨of space, we can finda countable set { x k } k ∈ N in X such that X = S k ∈ N U x k . Hence any D ∈ D n intersectswith some U x k and therefore D ∈ T ( x k ). Thus D n = S k ∈ N T ( x k ) is countable.Conversely, if X is a cosmic (respectively, an ℵ -space), then X is Lindel¨of (see [20])and it is trivially a σ -space (respectively, an ℵ -space). (cid:3) Corollary 5.10.
Let E be a Lindel¨of (in particular, separable metrizable) lcs. Then E is a weakly ℵ -space if and only if E is a weakly ℵ -space. Since every
W CG
Banach space is Lindel¨of in its weak topology by Preiss-Talagrand’stheorem (see [11, Theorem 12.35]), we note also
Corollary 5.11.
Every
W CG
Banach space is a weakly ℵ -space if and only if it is aweakly ℵ -space. As C c ( X ) is Lindel¨of for any ℵ -space X by [20, Proposition 10.3], we obtain Corollary 5.12.
Let X be an ℵ -space. Then C c ( X ) is a weakly ℵ -space if and onlyit is a weakly ℵ -space. Proof of Theorem 1.1
We need the following lemmas.
Lemma 6.1.
Let X be a completely regular space containing a non-scattered compactsubset K . Then C c ( X ) is not a weakly ℵ -space.Proof. Suppose for a contradiction that C c ( X ) is a weakly ℵ -space. As K is notscattered, there exists a continuous map f from K onto the interval [0 ,
1] (see [28,Theorem 8.5.4]). In particular, every compact subset of [0 ,
1] is the image f ( L ) for somecompact set L in X . By the Tietze-Urysohn theorem (which holds for compact subsetsof completely regular spaces, knowing that they have normal compactifications), f hasan extension ˜ f : X → [0 , f is also compact-covering, therefore the adjointmap h h ◦ ˜ f is an embedding of C [0 ,
1] into C c ( X ). Finally, if C [0 ,
1] were an ℵ -spacein the weak topology, then by Corollary 5.12 it would be an ℵ -space, which leads toa contradiction with the following result of Corson (see [20, Prop. 10.8]): A space ofthe form C ( K ) with K compact is an ℵ -space in the weak topology if and only if K is countable. (cid:3) For example, as X = N N has a non-scattered compact subset, the space C c ( X ) isan ℵ -space [20], but C c ( X ) is not a weakly ℵ -space by Lemma 6.1. Observe that thecondition on X to have only scattered (even countable) compact subsets is not enoughfor C c ( X ) to be a weakly ℵ -space. This follows from the following Lemma 6.2.
Let X be a Tychonoff space such that each compact subset of X is count-able. If C c ( X ) is a weakly ℵ -space, then X is separable. In particular, the space C c [0 , ω ) is not a weakly ℵ -space.Proof. By Proposition 3.4, there is a sequence { K n } n ∈ N of (countable) compact subsetsof X and a sequence { δ n } n ∈ N of positive numbers such that(6.1) \ n ∈ N { f ∈ C c ( X ) : f ( K n ) ⊆ [ − δ n , δ n ] } = { } . Set A := ∪ n ∈ N K n . Then A is countable. We show that A is dense in X . Indeed, if X \ cl X ( A ) = ∅ , we can find h = 0 such that h (cl X ( A )) = { } , that contradicts (6.1).The last assertion follows from the fact that [0 , ω ) is a non-separable locally compactnormal space (see [9, 3.1.27]). (cid:3) Lemma 6.3.
Let X be a completely regular space. Then the following assertions areequivalent: (i) X contains a non-scattered compact subset. (ii) C c ( X ) contains a copy of ℓ . (iii) C c ( X ) contains a separable Banach space B with non-separable dual.So, every compact subset of X is scattered if and only if C c ( X ) does not contain a copyof ℓ .Proof. (i) ⇒ (ii) Assume that X contains a non-scattered compact set K . As it wasshown in the proof of Lemma 6.1, the space C [0 ,
1] embeds into C c ( X ). It remains tonote that C [0 ,
1] contains ℓ .(ii) ⇒ (iii) is trivial. Let us prove that (iii) ⇒ (i) Suppose for a contradiction thatevery compact subset of X is scattered. Denote by K the set of all compact subsets of X . Then C c ( X ) can be treated as a subspace of the product E := Q K ∈K C c ( K ). Then,by [8, Theorem 4.1 and the first claim of the proof], the space B is embedded in thefinite product F k := C ( K i ) × · · · × C ( K i k ) for some k ∈ N . On the other hand, since F k = C ( K i ⊕ · · · ⊕ K i k ) and the topological direct sum K := K i ⊕ · · · ⊕ K i k is compactand scattered, F k is Asplund by [11, Theorem 12.29] (i.e. every separable subspace of C ( K ) has separable dual). Thus F k does not contain B , a contradiction. (cid:3) We recall that the countable product of ℵ -spaces is an ℵ -space (see [20]). Now weare ready to prove the main theorem. Proof of Theorem 1.1.
Let ( K n ) n be a fundamental sequence of compact sets in X .Then C c ( X ) is embedded in the product Q n C ( K n ).If X is countable, each space K n is metrizable and scattered, so C c ( X ) is a weakly ℵ -space by Proposition 5.2(i). Thus C c ( X ) is a weakly ℵ -space.Assume that C c ( X ) is a weakly ℵ -space. By Lemma 6.1, K n is scattered for every n ∈ N . We apply Lemma 6.3 to derive that C c ( X ) does not contain a copy of ℓ . NowCorollary 5.8 says that C c ( X ) is a weakly ℵ -space.Assume now that C c ( X ) is a weakly ℵ -space. Lemma 6.1 shows that every compactsubset of X is scattered. Since C c ( X ) is separable, X admits a weaker metrizabletopology; so all sets K n are metrizable and scattered. Thus each K n is countable, soit is the whole space X . (cid:3) ETWORKS FOR THE WEAK TOPOLOGY 0F BANACH AND FR´ECHET SPACES 13
Theorem 1.1 combined with [19] provides concrete Banach spaces C ( K ) which underthe weak topology are σ -spaces but is not ℵ -spaces. Corollary 6.4.
Let K be an uncountable separable compact space. If K is(1) a linearly ordered space, or(2) a dyadic space,then C ( K ) endowed with the weak topology is a σ -space but not an ℵ -space. Ifadditionally K is metrizable, then C ( K ) endowed with the weak topology is a cosmicspace but is not an ℵ -space.Proof. For the both cases, by Theorem 1.1, the space C ( K ) with the weak topology isnot an ℵ -space.(1): Let K be a compact space as assumed. By [19, Theorem 5.5] the space( C ( K ) , τ p ) is a σ -space, where τ p is the pointwise topology on C ( K ). Moreover, byLemma 5.4 and Lemma 2.3.1 of [19] the space C ( K ) admits a σ -discrete collectionin ( C ( K ) , τ p ) which is a network in C ( K ). Hence C ( K ) with the weak topology is a σ -space.(2): By [19, Lemma 2.3.1 and Lemma 5.10] there exists a σ -discrete family in( C (2 ω ) , τ p ) which is a network in C (2 ω ). Hence the space C (2 ω ) endowed with theweak topology is a σ -space. Since K is a continuous image of 2 ω , the space C ( K )embeds into C (2 ω ) for the weak topology. This proves the general case.Assume now that K is metrizable. Then C ( K ) is a Polish space. So C ( K ) endowedwith the weak topology is a cosmic space by [20] but is not an ℵ -space by Theorem1.1. (cid:3) Remark 6.5.
Let K be a compact space. Then C ( K ) is weakly cosmic if and onlyif C ( K ) is separable if and only if K is metrizable. So, if K satisfies (1) or (2) ofCorollary 6.4 but is not metrizable (for example, K is the Bohr compactification of acountably infinite abelian group; in this case K satisfies (2) but is not metrizable) weobtain a non-trivial example of Banach spaces B such that ( B, σ ( B, B ′ )) is a σ -spacebut ( B, σ ( B, B ′ )) is neither cosmic nor an ℵ -space. Corollary 6.6.
Let X be a locally compact and paracompact space. Then C c ( X ) is aweakly ℵ -space if and only if X is countable.Proof. By the assumption on X , there exists a family { X t : t ∈ T } of locally com-pact and σ -compact spaces such that X := L t ∈ T X t , see [9, 5.1.27]. So C c ( X ) = Q t ∈ T C c ( X t ) and each C c ( X t ) is a Fr´echet space.Assume that C c ( X ) is a weakly ℵ -space. Then C c ( X t ) is a weakly ℵ -space andhence X t is countable by Theorem 1.1 for every t ∈ T . As any compact subset of C c ( X ) is metrizable by Proposition 3.4(i), the set T is countable. Thus X is countable.Conversely, if X is countable, C c ( X ) is a Fr´echet space and Theorem 1.1 applies. (cid:3) Since ℓ ∞ = C ( β N ), Theorem 1.1 provides Corollary 6.7.
A Banach space containing a copy of ℓ ∞ is not a weakly ℵ -space. Corollary 6.8.
Let X be a locally compact Hausdorff space. Then the Banach space C ( X ) of continuous functions on X vanishing at infinity is a weakly ℵ -space if andonly if X is countable. Proof.
Let K be the one-point compactification of X . Then C ( K ) = C ( X ) ⊕ R ,therefore Theorem 1.1 applies. (cid:3) Question 6.9.
Let X be a Tychonoff space. Is it true that X is countable providedthat C c ( X ) is a weakly ℵ -space? C c ( X ) over a countable ℵ -space X for the weak topology This section is motivated by the previous one; especially by Question 6.9. Let µ bea ( σ -additive real-valued regular) measure on a Tychonoff space X . The variation andthe norm of µ are denoted by | µ | and k µ k respectively. We shall use the following wellknown fact (see for example [17, 7.6.5]). Fact 7.1.
Let X be a Tychonoff space. Then (i) ( C c ( X )) ′ can be identified with the space of all measures on X with compactsupport. (ii) ( C p ( X )) ′ can be identified with the space of measures with finite support in X . If X is a countable ℵ -space, the space C ( X ) is an ℵ -space both in the compact-open and the pointwise topology (see [20]). Next theorem shows that the same holdsalso for the weak topology on C c ( X ). Theorem 7.2. If X is a countable ℵ -space, then C c ( X ) is a weakly ℵ -space.Proof. Set E := C c ( X ) and let E w be the space C c ( X ) endowed with the weak topology.Let D be a countable closed k -network in X closed under taking finite unions, and let B be a countable basis in R . For every finite subset F = { x , . . . , x n } of X , every finitesubfamily U = { U , . . . , U n } of B , each D ∈ D and every m ∈ N , set(7.1) A ( F, U , D, m ) := { f ∈ C ( X ) : f ( x i ) ∈ U i , ≤ i ≤ n, and f ( D ) ⊂ [ − m, m ] } . Denote by A the countable family of all subsets of E of the form (7.1). By [16, Thm.1], in order to prove the theorem it is enough to show that the family A satisfies thefollowing claim. Claim . For every f ∈ E , for every sequence { f n } n ∈ N converging to f in E w andany neighborhood W of f in E w there exists A ∈ A such that f ∈ A ⊂ W and f n ∈ A for almost all n ∈ N . Without loss of generality we may assume that W is of the standard form, i.e. thereare measures µ , . . . , µ s ∈ E ′ and ε > W = { f ∈ E : | µ i ( f − f ) | < ε, ≤ i ≤ s } . Set K := S si =1 supp( µ i ). So K is a compact subset of X by Fact 7.1(i).Let { D ′ n } n ∈ N be an enumeration of the family { D ′ ∈ D : K ⊆ D ′ } . For every n ∈ N ,set D n := T ni =1 D ′ i . It follows that the decreasing sequence of sets { D n } n ∈ N convergesto the compact set K in the sense that each neighborhood O ( K ) of K contains all butfinitely many sets D n . Step 1 . Let us show that there are k, m ∈ N such that(7.2) | f i ( x ) | < m, ∀ x ∈ D m , ∀ i ≥ k. ETWORKS FOR THE WEAK TOPOLOGY 0F BANACH AND FR´ECHET SPACES 15
Indeed, assuming the converse we choose a sequence { x n } n ∈ N , with x n ∈ D n for every n ∈ N , and a sequence i < i < . . . such that(7.3) | f i n ( x n ) | > n, ∀ n ∈ N . Since { D n } n ∈ N converges to the compact set K , all accumulation points of the sequence { x n } n ∈ N are in K . In other words, the set K ′ := K ∪ { x n } n ∈ N is compact. As the restriction map f f | K ′ is continuous, we obtain that the sequence S := { f i n | K ′ } n ∈ N in the Banach space C ( K ′ ) converges to f | K ′ in the weak topologyof C ( K ′ ). Thus S is bounded, that is there is C > | f i n ( x ) | < C, ∀ x ∈ K ′ , ∀ n ∈ N . In particular, | f i n ( x n ) | < C for every n ∈ N , that contradicts (7.3). This proves (7.2). Step 2 . Fix k, m ∈ N such that (7.2) holds. For every 1 ≤ i ≤ s take a finite subset F i of supp( µ i ) such that(7.4) | µ i | (supp( µ i ) \ F i ) < ε m , and for every x i,j ∈ F i choose U i,j ∈ B such that(7.5) f ( x i,j ) ∈ U i,j and diam( U i,j ) < ε | F i | · k µ i k . If x i,j = x k,l for some 1 ≤ i < k ≤ s , we shall suppose that U i,j = U k,l . Finally we set A := { f ∈ C ( X ) : f ( x i,j ) ∈ U i,j , x i,j ∈ F i , ≤ i ≤ s ; f ( D m ) ⊂ [ − m, m ] } . Clearly, A ∈ A . For each f ∈ A and every 1 ≤ i ≤ s , (7.4) and (7.5) imply | µ i ( f − f ) | ≤ X x i,j ∈ F i | f ( x i,j ) − f ( x i,j ) | · k µ i k + X x i,j ∈ supp( µ i ) \ F i | f ( x i,j ) − f ( x i,j ) | · | µ i ( { x i,j } ) | < | F i | · ε | F i | · k µ i k · k µ i k + 2 m · ε m = ε. Thus A ⊂ W , and (7.2) shows that f i ( D m ) ⊂ [ − m, m ] for every i ≥ k . Since f n → f also in the pointwise topology, we obtain that f n ∈ A for all sufficiently large n ∈ N .This proves Claim and hence also the theorem. (cid:3) Consequently, the space C c ( X ) is a weakly ℵ -space for any metrizable and countablespace X . We end this section with the following conjecture Conjecture 7.3.
Let X be a Tychonoff space. Then C c ( X ) is a weakly ℵ -space if andonly if C c ( X ) is a weakly ℵ -space if and only if X is a countable ℵ -space. One application
Let E be a separable Banach space and S its closed unit ball endowed with the weaktopology of E . In [10, Theorem A] Edgar and Wheller proved that S is completelymetrizable if and only if S is a Polish space if and only if S is metrizable and everyclosed subset of S is a Baire space. We supplement this result.For this purpose we introduce a property stronger than the property to be an ℵ -space by [20, Theorem 11.4]. We say that a topological space X is an ℵ -space if X is a continuous image under a compact-covering map from a Polish space Y . Everyclosed subspace of an ℵ -space is also an ℵ -space. Proposition 8.1.
Let E be a separable Banach space. (i) If E does not contain a copy of ℓ , then the closed unit ball S of E is a Polishspace in the weak topology of E if and only if E is a weakly ℵ -space. (ii) If E contains a copy of c , then E is not a weakly ℵ -space.Proof. (i) The closed unit ball S endowed with the weak topology we denote by S w .Assume that S w is a Polish space. For each n ∈ N set S n := nS w and let Y := L n S n be the topological direct sum of the sequence ( S n ) n of Polish spaces. Denote by T thecanonical mapping from Y onto E w := ( E, σ ( E, E ′ )). Since every compact set of E w is contained in some S m , the map T is continuous and compact-covering. Conversely,assume that E is a weakly ℵ -space and T : Y → E w is a continuous compact-coveringmap. Denote by B ( x, r ) the closed ball in Y of radius r centered at x . For a countabledense sequence ( x j ) j ∈ N in Y and each α = ( n k ) ∈ N N , set K α := T ∞ k =1 S n k j =1 B ( x j , k − ).Then { K α : α ∈ N N } is a family of compact sets in Y covering Y with K α ⊂ K β whenever α ≤ β , and such that every compact set in Y is contained in some K α .Set W α := T ( K α ) for each α ∈ N N . Since T is compact-covering and continuous, thesets W α compose a compact covering of E w such that every σ ( E, E ′ )-compact set iscontained in some W α . On the other hand, E w is an ℵ -space, so we apply Corollary5.6 to deduce that the strong dual E ′ is separable. Hence S w is metrizable (see Lemma5.1) and separable. Now by Christensen’s theorem, see [18, Theorem 6.1], the space S w is a Polish space.(ii) The closed unit ball B of c is metrizable and separable in the weak topology.On the other hand, by [10, Theorem A, Examples (3)] B is not a Polish space in theweak topology. Now the proof of (i) (involving the Christensen’s theorem) applies tocomplete the case (ii). (cid:3) Remark 8.2.
Note that ℓ is a weakly ℵ -space (by the Schur property) but the unitball in ℓ is not a Polish space, see [10, Example 9]. So the assumption on E in item (i)that E does not contain a copy of ℓ is essential. Recall also that for a Banach space E with separable bidual E ′′ the unit ball S in E is a Polish space in the weak topologyby Godefroy’s theorem, see [11, Theorem 12.55]. Remark 8.3.
Let K be a countably infinite compact space. Then C ( K ) contains acopy of c . Hence C ( K ) is not a weakly ℵ -space by (ii), but C ( K ) is a weakly ℵ -spaceby Theorem 1.1. References
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Department of Mathematics, Ben-Gurion University of the Negev, Beer-Sheva P.O.653, Israel
E-mail address : [email protected] Faculty of Mathematics and Informatics A. Mickiewicz University, − Pozna´n,Poland.
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