ω ω -Base and infinite-dimensional compact sets in locally convex spaces
aa r X i v : . [ m a t h . GN ] J u l ω ω -BASE AND INFINITE-DIMENSIONAL COMPACT SETS INLOCALLY CONVEX SPACES TARAS BANAKH, JERZY KA¸ KOL, AND JOHANNES PHILLIP SCH ¨URZ
Abstract.
A locally convex space (lcs) E is said to have an ω ω -base if E has a neigh-borhood base { U α : α ∈ ω ω } at zero such that U β ⊆ U α for all α ≤ β . The class of lcswith an ω ω -base is large, among others contains all ( LM )-spaces (hence ( LF )-spaces),strong duals of distinguished Fr´echet lcs (hence spaces of distributions D ′ (Ω)). A remark-able result of Cascales-Orihuela states that every compact set in a lcs with an ω ω -baseis metrizable. Our main result shows that every uncountable-dimensional lcs with an ω ω -base contains an infinite-dimensional metrizable compact subset. On the other hand,the countable-dimensional space ϕ endowed with the finest locally convex topology hasan ω ω -base but contains no infinite-dimensional compact subsets. It turns out that ϕ is a unique infinite-dimensional locally convex space which is a k R -space containing noinfinite-dimensional compact subsets. Applications to spaces C p ( X ) are provided. Introduction
A topological space X is said to have a neighborhood ω ω -base at a point x ∈ X if thereexists a neighborhood base ( U α ( x )) α ∈ ω ω at x such that U β ( x ) ⊆ U α ( x ) for all α ≤ β in ω ω . We say that X has an ω ω -base if it has a neighborhood ω ω -base at each point of X .Evidently, a topological group (particularly topological vector space (tvs)) has an ω ω -baseif it has a neighborhood ω ω -base at the identity. The classical metrization theorem ofBirkhoff and Kakutani states that a topological group G is metrizable if and only if G is first-countable. Then, as easily seen, if ( U n ) n ∈ ω is a neighborgood base at the identityof G , then the family { U α : α ∈ ω ω } formed by sets U α = U α (0) forms an ω ω -base (atthe identity) for G . Locally convex spaces (lcs) with an ω ω -base are known in FunctionalAnalysis since 2003 when Cascales, K¸akol, and Saxon [7] characterized quasi-barreled lcswith an ω ω -base. In several papers (see [16] and the references therein) spaces with an ω ω -base were studied under the name lcs with a G -base , but here we prefer (as in [4]) touse the more self-suggesting terminology of ω ω -bases.In [8] Cascales and Orihuela proved that compact subsets of any lcs with an ω ω -baseare metrizable. This refers, among others, to each ( LM )-space, i.e. a countable inductivelimit of metrizble lcs, since ( LM )-spaces have an ω ω -base. Also the following metrizationtheorem holds together a number of topological conditions. Theorem 1.1. [16, Corollary 15.5]
For a barrelled lcs E with an ω ω -base, the followingconditions are equivalent. The research for the second named author is supported by the GA ˇCR project 20-22230L and RVO:67985840. (1) E is metrizable; (2) E is Fr´echet-Urysohn; (3) E is Baire-like; (4) E does not contain a copy of ϕ , i.e. an ℵ -dimensional vector space endowed withthe finest locally convex topology. Hence every Baire lcs with an ω ω -base is metrizable. The space ϕ appearing in Theo-rem 1.1 has the following properties:(1) ϕ is the strong dual of the Fr´echet-Schwartz space R ω .(2) All compact subsets in ϕ are finite-dimensional.(3) ϕ is a complete bornological space,see [23], [21], [16].Being motivated by above’s results, especially by a remarkable theorem of Cascales-Oruhuela mentioned above, one can ask for a possible large class of lcs E for which everyinfinite-dimensional subspace of E contains an infinite-dimensional compact (metrizable)subset. Surely, each metrizable lcs trivially fulfills this request. We prove however thefollowing general Theorem 1.2.
Every uncountably-dimensional lcs E with ω ω -base contains an infinite-dimensional metrizable compact subset. Theorem 1.2 will be proved in Section 4. An alternative proof will be presented inSection 5 as a consequence of Theorem 5.2.The uncountable dimensionality of the space E in Theorem 1.2 cannot be replaced bythe infinite-dimensionality of E : the space ϕ is infinite-dimensional, has an ω ω -base andcontains no infinite-dimensional compact subsets. However, ϕ is a unique locally convex k R -space with this property. Recall [20] that a topological space X is a k R -space if a function f : X → R is continuous whenever for every compact subset K ⊆ X the restriction f ↾ K is continuous. We prove the following Theorem 1.3.
A lcs E is topologically isomorphic to the space ϕ if and only if E is a k R -space containing no infinite-dimensional compact subsets. Theorem 1.3 implies that a lcs is topologically isomorphic to ϕ if and only if it is home-omorphic to ϕ . This topological uniqueness property of the space ϕ was first proved bythe first author in [2].The following characterization of the space ϕ can be derived from Theorems 1.2 and 2.1.It shows that ϕ is a unique bornological space for which the uncountable dimensionalityin Theorem 1.2 cannot be weakened to infinite dimensionality. Theorem 1.4.
A lcs E is topologically isomorphic to the space ϕ if and only if E isbornological, has an ω ω -base and contains no infinite-dimensional compact subset. Theorem 1.2 provides a large class of concrete (non-metrizable) lcs containing infinite-dimensional compact sets.
Corollary 1.5.
Every uncountable-dimensional subspace of an ( LM ) -space contains aninfinite-dimensional compact set. ω -BASE AND INFINITE-DIMENSIONAL COMPACT SETS IN LOCALLY CONVEX SPACES 3 Let X be a Tychonoff space. By C p ( X ) and C k ( X ) we denote the space of continuousreal-valued functions on X endowed with the pointwise and the compact-open topology,respectively. The problem of characterization of Tychonoff spaces X whose function spaces C p ( X ) and C k ( X ) admit an ω ω -base is already solved. Indeed, by [16, Corollary 15.2] C p ( X )has an ω ω -base if and only if X is countable. The space C k ( X ) has an ω ω -base if and onlyif X admits a fundamental compact resolution [11], for necessary definitions see below.Since every ˇCech-complete Lindel¨of space X is a continuous image of a Polish space undera perfect map (and the latter space admits a fundamental compact resolution), the space C p ( X ) has an ω ω -base. So, we have another concrete application of Theorem 1.2. Example 1.6.
Let X be an infinite ˇCech-complete Lindel¨of space. Then every uncountable-dimensional subspace of C k ( X ) contains an infinite-dimensional metrizable compact set.In Section 2 we show that all (bornological) lcs containing no infinite-dimensional com-pact subsets are bornologically (and topologically) isomorphic to a free lcs over discretetopological spaces. Consequently, in Sections 3 and 4 we study the free lcs L ( κ ) over infi-nite cardinals κ , including L ( ω ) = ϕ . We introduce two concepts: the ( κ, λ )-tall bornologyand the ( κ, λ ) p -equiconvergence, which will be used to obtain bornological and topologicalcharacterizations of L ( κ ). Both concepts apply to prove Theorem 1.2. To this end, we shallprove that each topological (vector) space with an ω ω -base is ( ω , ω ) p -equiconvergent (andhas ( ω , ω )-tall bornology). Another property implying the ( ω , ω ) p -equiconvergence is theexistence of a countable cs • -network (see Theorem 4.2), which follows from the existenceof an ω ω -base according to Proposition 3.3. Linear counterparts of cs • -networks are radialnetworks introduced in Section 5, whose main result is Theorem 5.2 implying Theorem 1.2.Some applications of Theorem 1.2 to function spaces C p ( X ) are provided in Section 6.2. Locally convex spaces containing no infinite-dimensional compactsubsets
In this section we study lcs containing no infinite-dimensional compact subsets. We shallshow that all such (bornological) spaces are bornologically (and topologically) isomorphicto free lcs over discrete topological spaces.Recall that for a topological space X its free locally convex space is a lcs L ( X ) endowedwith a continuous function δ : X → L ( X ) such that for any continuous function f : X → E to a lcs E there exists a unique linear continuous map T : L ( X ) → E such that T ◦ δ = f .The set X forms a Hamel basis for L ( X ) and δ is a topological embedding, see [22]; we alsorefer to [5] and [4] for several results and references concerning this concept; [5, Theorem5.4] characterizes those X for which L ( X ) has an ω ω -base.Let E be a tvs. A subset B ⊆ E is called bounded if for every neighborhood U ⊆ E ofzero there exists n ∈ N such that B ⊆ nU . The family of all bounded sets of E is calledthe bornology of E . A linear operator f : E → F between two tvs is called bounded if forany bounded set B ⊆ E its image f ( B ) is bounded in F .Two tvs E and F are • topologically isomorphic if there exists a linear bijective function f : E → F suchthat f and f − are continuous; TARAS BANAKH, JERZY KA¸ KOL, AND JOHANNES PHILLIP SCH ¨URZ • bornologically isomorphic if there exists a linear bijective function f : E → F suchthat f and f − are bounded.A lcs E is called bornological if each bounded linear operator from E to a lcs F is continuous.A linear space E is called κ -dimensional if E has a Hamel basis of cardinality κ . In thiscase we write κ = dim( E ).A lcs E is free if it carries the finest locally convex topology. In this case E is topologicallyisomorphic to the free lcs L ( κ ) over the cardinal κ = dim( E ) endowed with the discretetopology.The study around the free lcs L ( ω ) = ϕ has attracted specialists for a long time. Forexample, Nyikos observed [21] that each sequentially closed subset of L ( ω ) is closed al-though the sequential closure of a subset of ϕ need not be closed. Consequently, L ( ω )is a concrete “small” space without the Fr´echet-Urysohn property. Applying the Bairecategory theorem one shows that L ( ω ) is not a Baire-like space (in sense of Saxon [23])and a barrelled lcs E is Baire-like if E does not contain a copy of L ( ω ), see [23]. Although L ( ω ) is not Fr´echet-Urysohn, it provides some extra properties since all vector subspacesin L ( ω ) are closed. In [17] we introduced the property for a lcs E (under the name C − )stating that the sequential closure of every linear subspace of E is sequentially closed,and we proved [17, Corollary 6.4] that the only infinite-dimensional Montel (DF)-spacewith property C − is L ( ω ) (yielding a remarkable result of Bonet and Defant that the onlyinfinite-dimensional Silva space with property C − is L ( ω )). This implies that barrelled( DF )-spaces and ( LF )-spaces satisfying property C − are exactly of the form M , L ( ω ), or M × L ( ω ) where M is metrizable, [17, Theorems 6.11, 6.13].The following simple theorem characterizes lcs containing no infinite-dimensional com-pact subsets. Theorem 2.1.
For a lcs E the following conditions are equivalent: (1) Each compact subset of E has finite topological dimension. (2) Each bounded linearly independent set in E is finite. (3) E is bornologically isomorphic to a free lcs.If E is bornological, then the conditions (1)–(3) are equivalent to (4) E is free.Proof. (1) ⇒ (2) Suppose that each compact subset of E has finite topological dimen-sion. Assuming that E contains an infinite bounded linearly indendent set, we can find abounded linearly independent set { x n } n ∈ ω consisting of pairwise distinct points x n . Thenthe sequence (2 − n x n ) n ∈ ω converges to zero and K = S n ∈ ω (cid:8) n P k = n t k x k : ( t k ) nk = n ∈ n Q k = n [0 , − k ] (cid:9) is an infinite-dimensional compact set in E , which contradicts our assumption.(2) ⇒ (3) Let τ be the finest locally convex topology on E . Then the identity map( E, τ ) → E is continuous and hence bounded. If each bounded linearly independent set in E is finite, then each bounded set B ⊆ E is contained in a finite-dimensional subspace of ω -BASE AND INFINITE-DIMENSIONAL COMPACT SETS IN LOCALLY CONVEX SPACES 5 E and hence is bounded in the topology τ . This means that the identity map E → ( E, τ )is bounded and hence E is bornologically isomorphic to the free lcs ( E, τ ).(3) ⇒ (1) If E is bornologically isomorphic to a free lcs F then each bounded linearlyindependent set in E is finite, since the free lcs F has this property.The implication (4) ⇒ (3) is trivial. If E is bornological then the implication (3) ⇒ (4)follows from the continuity of bounded linear operators on bornological spaces. (cid:3) The free lcs over discrete topological spaces are not unique lcs possessing no infinite-dimensional compact sets. A subset B of a topological space X is called functionallybounded if for any continuous real-valued function f : X → R the set f ( B ) is bounded. Proposition 2.2.
For a Tychonoff space X the following conditions are equivalent: (1) each compact subset of the free lcs L ( X ) has finite topological dimension; (2) each bounded linearly independent set in L ( X ) is finite; (3) each functionally bounded subset of X is finite.Proof. The equivalence (1) ⇔ (2) follows from the corresponding equivalence in Theo-rem 2.1. The implication (3) ⇒ (1) follows from [6, Lemma 10.11.3], and (2) ⇒ (3) followsfrom the observation that each functionally bounded set in a lcs is bounded. (cid:3) Bornological and topological characterizations of the spaces L ( κ )In this section, given an infinite cardinal κ we characterize the free lcs L ( κ ) using somespecific properties of the bornology and the topology of the space L ( κ ).Let κ, λ be two cardinals. A lcs E is defined to have ( κ, λ ) -tall bornology if every subset A ⊆ E of cardinality | A | = κ contains a bounded subset B ⊆ A of cardinality | A | = λ . Theorem 3.1.
Let κ be an infinite cardinal. For a lcs E the following conditions areequivalent: (1) E is bornologically isomorphic to the free lcs L ( κ ) ; (2) each bounded linearly independent set in E is finite and the bornology of E is ( κ + , ω ) -tall but not ( κ, ω ) -tall.If E is bornological, then the conditions (1)–(2) are equivalent to (3) E is topologically isomorphic to L ( κ ) .Proof. (1) ⇒ (2): Assume that E is bornologically isomorphic to L ( κ ). Then E hasalgebraic dimension κ and each bounded linearly independent set in E is finite (since thisis true in L ( κ )).To see that the bornology of E is ( κ + , ω )-tall, take any set K ⊆ E of cardinality | K | = κ + . Since E has algebraic dimension κ , there exists a cover ( B α ) α ∈ κ of E by κ manycompact sets. By the Pigeonhole Principle, there exists α ∈ κ such that | K ∩ B α | = κ + .This means that the bornology of E is ( κ + , κ + )-tall and hence ( κ + , ω )-tall.To see that the bornology of the space E is not ( κ, ω )-tall, observe that the Hamelbasis κ of L ( κ ) has the property that no infinite subset of κ is bounded in L ( κ ). Since E is bornologically isomorphic to L ( κ ), the image of κ in E is a subset of cardinality κ containing no bounded infinite subsets and witnessing that E is not ( κ, ω )-tall. TARAS BANAKH, JERZY KA¸ KOL, AND JOHANNES PHILLIP SCH ¨URZ (2) ⇒ (1): Assume that each bounded linearly independent set in E is finite and thebornology of E is ( κ + , ω )-tall but not ( κ, ω )-tall. Let B be a Hamel basis of E . We claimthat | B | = κ . Assuming that | B | > κ , we conclude that E is not ( κ + , ω )-tall, which isa contradiction. Assuming that | B | < κ , we conclude that E is the union of < κ manybounded sets and hence is ( κ, κ )-tall by the Pigeonhole Principle. But this contradicts ourassumption. Therefore | B | = κ . Let h : κ → B be any bijection and ¯ h : L ( κ ) → E be theunique extension of h to a linear continuous operator. Since B is a Hamel basis for E , theoperator ¯ h is bijective. Since each bounded set in E is contained in a finite-dimensionallinear subspace, the operator ¯ h − : E → L ( κ ) is bounded and hence ¯ h : L ( κ ) → E is abornological isomorphism.If the space E is bornological, then the equivalence (1) ⇔ (3) follows from the bornolog-ical property of E and L ( κ ). (cid:3) The ( κ, ω )-tallness of the bornology of a lcs E has topological counterparts introducedin the following definition. Definition 3.2.
Let κ, λ be cardinals. We say that a topological space X is • ( κ, λ ) p -equiconvergent at a point x ∈ X if for any indexed family { x α } α ∈ κ ⊆ { s ∈ X ω : lim n →∞ s ( n ) = x } , there exists a subset Λ ⊆ κ of cardinality | Λ | = λ suchthat for every neighborhood O x ⊆ X of x there exists n ∈ ω such that the set { α ∈ Λ : x α ( n ) / ∈ O x } is finite; • ( κ, λ ) k -equiconvergent at a point x ∈ X if for any indexed family { x α } α ∈ κ ⊆ { s ∈ X ω : lim n →∞ s ( n ) = x } , there exists a subset Λ ⊆ κ of cardinality | Λ | = λ suchthat for every neighborhood O x ⊆ X of x there exists n ∈ ω such that for every m ≥ n and α ∈ Λ we have x α ( m ) ∈ O x ; • ( κ, λ ) p -equiconvergent if X is ( κ, λ ) p -equiconvergent at every point x ∈ X ; • ( κ, λ ) k -equiconvergent if X is ( κ, λ ) k -equiconvergent at every point x ∈ X .It is easy to see that every ( κ, λ ) k -equiconvergent space is ( κ, λ ) p -equiconvergent. Thefollowing observation will be used below. Proposition 3.3.
If a lcs E is ( κ, λ ) p -equiconvergent, then its bornology is ( κ, λ ) -tall.Proof. Given a subset K ⊆ E of cardinality | K | = κ , for every α ∈ K consider theconvergent sequence x α ∈ X ω defined by x α ( n ) = 2 − n α . Assuming that the lcs E is( κ, λ ) p -equiconvergent, we can find a subset L ⊆ K of cardinality | L | = λ such that forevery neighborhood of zero U ⊆ E there exists n ∈ ω such that the set { α ∈ L : 2 − n α / ∈ U } is finite. We claim that the set L is bounded. Indeed, for every neighborhood U ⊆ E ofzero, we find a neighborhood V ⊆ E of zero such that [0 , · V ⊆ U . By our assumption,there exists n ∈ ω such that the set F = { α ∈ K : 2 − n α / ∈ V } is finite. Find m ≥ n suchthat 2 − m α ∈ U for every α ∈ F . Then 2 − m L ⊆ − m ( L \ F ) ∪ − m F ⊆ ([0 , · V ) ∪ U = U ,and hence the set L is bounded. (cid:3) Nevertheless, it seems that the following question remains open.
Problem 3.4.
Assume that the bornology of a lcs E is ( ω , ω ) -tall. Is it true that E is ( ω , ω ) p -equiconvergent? ω -BASE AND INFINITE-DIMENSIONAL COMPACT SETS IN LOCALLY CONVEX SPACES 7 Below we prove the following topological counterpart to Theorem 3.1.
Theorem 3.5.
Let κ be an infinite cardinal. For a lcs E the following conditions areequivalent: (1) E is bornologically isomorphic to L ( κ ) ; (2) each compact subset of E has finite topological dimension, E is ( κ + , ω ) k -equiconvergentbut not ( κ, ω ) p -equiconvergent. (3) each compact subset of E has finite topological dimension, E is ( κ + , ω ) p -equiconvergentbut not ( κ, ω ) k -equiconvergent.If E is bornological, then the conditions (1)–(3) are equivalent to (4) E is topologically isomorphic to L ( κ ) .Proof. (1) ⇒ (2): Assume that E is bornologically isomorphic to L ( κ ). By Theorems 3.1each bounded linearly independent set in E is finite, and by Theorem 2.1, each compactsubset of E is finite-dimensional. The linear space E has algebraic dimension κ , beingisomorphic to the linear space L ( κ ). Let B be a Hamel basis for the space E .To show that E is ( κ + , ω ) k -equiconvergent, fix an indexed family { x α } α ∈ κ + ⊆ { s ∈ E ω :lim n →∞ s ( n ) = 0 } . Since bounded linearly independent sets in E are finite, for every α ∈ κ + there exists a finite set F α ⊆ B such that the bounded set x α [ ω ] is contained in the linearhull of F α . Since | B | = κ < κ + , by the Pigeonhole Principle, for some finite set F ⊆ B theset A = { α ∈ κ + : F α = F } is uncountable. Let [ F ] be the linear hull of the finite set F inthe linear space E .Consider the ordinal ω + 1 = ω ∪ { ω } endowed with the compact metrizable topologygenerated by the linear order. For every α ∈ A let ¯ x α : ω + 1 → [ F ] be the continuousfunction such that ¯ x α ↾ ω = x α and ¯ x α ( ω ) = 0. Let C k ( ω + 1 , [ F ]) be the space of con-tinuous functions from ω + 1 to [ F ], endowed with the compact-open topology. Since A is uncountable and the space C k ( ω + 1 , [ F ]) ⊇ { ¯ x α } α ∈ A is Polish, there exists a sequence { α n } n ∈ ω ⊆ A of pairwise distinct ordinals such that the sequence (¯ x α n ) n ∈ ω converges to¯ x α in the function space C k ( ω + 1 , [ F ]). Then the set Λ = { α n } n ∈ ω ⊆ κ + witnesses that E is ( κ + , ω ) k -equiconvergent to zero and by the topological homogeneity, E is ( κ + , ω )-equiconvergent. By Theorem 3.1, the bornology of the space E is not ( κ, ω )-tall. ByProposition 3.3, the space E is not ( κ, ω ) p -equiconvergent.The implication (2) ⇒ (3) is trivial. To prove that (3) ⇒ (1), assume that each compactsubset of E has finite topological dimension and E is ( κ + , ω ) p -equiconvergent but not( κ, ω ) k -equiconvergent. Let B be a Hamel basis in E . By Theorem 2.1, the space E isbornologically isomorphic to L ( | B | ). Applying the (already proved) implication (1) ⇒ (2),we conclude that E is ( | B | + , ω ) k -equiconvergent, which implies that | B | ≥ κ (as E is not( κ, ω ) k -equiconvergent). Assuming that | B | > κ , we can see that the family { x b } b ∈ B ⊆ E ω of the sequences x b ( n ) = 2 − n b witnesses that E is not ( | B | , ω ) p -equiconvergent and hencenot ( κ + , ω ) p -equiconvergent, which contradicts our assumption. So, | B | = κ and E isbornologically isomorphic to L ( κ ). If the space E is bornological, then the equivalence(1) ⇔ (4) follows from the bornological property of E and L ( κ ). (cid:3) TARAS BANAKH, JERZY KA¸ KOL, AND JOHANNES PHILLIP SCH ¨URZ
Observe that the purely topological properties (2), (3) in Theorem 3.5 characterize thefree lcs L ( κ ) up to bornological equivalence. We do not know whether the topologicalstructure of the space L ( κ ) determines this lcs uniquely up to a topological isomorphism. Problem 3.6.
Assume that a lcs E is homeomorphic to the free lcs L ( κ ) for some cardinal κ . Is E topologically isomorphic to L ( κ ) ? By [2] the answer to this problem is affirmative for κ = ω . This affirmative answer canalso be derived from the following topological characterizations of the space L ( ω ) = ϕ .This characterization has been announced in the introduction as Theorem 1.3. Theorem 3.7.
A lcs E is topologically isomorphic to the free lcs L ( ω ) if and only if E isan infinite-dimensional k R -space containing no infinite-dimensional compact subset.Proof. The “only if” part follows from known topological properties of the space L ( ω ) = ϕ mentioned in the introduction. To prove the “if” part, assume that a lcs E is a k R -space and each compact subset of E is finite-dimensional. Choose a Hamel basis B in E and consider the linear continuous operator T : L ( B ) → E such that T ( b ) = b for each b ∈ B . Since B is a Hamel basis, the operator T is injective. We claim that the operator T − : E → L ( B ) is bounded. By Theorem 2.1 the linear hull of each compact subset K ⊆ E is finite-dimensional, which implies that the restriction T − ↾ K is continuous. Since E is a k R -space, T − is continuous and hence T is a topological isomorphism. Then thefree lcs L ( B ) is a k R -space. Applying [15], we conclude that B is countable and hence E is topologically isomorphic to L ( ω ). (cid:3) A Tychonoff space X is called Ascoli if the canonical map δ : X → C k ( C k ( X )) assigningto each point x ∈ X the Dirac functional δ x : C k ( X ) → R , δ x : f f ( x ), is continuous.By [3], the class of Ascoli spaces includes all Tychonoff k R -spaces. By [15] a Tychonoffspace X is countable and discrete if and only if its free lcs L ( X ) is Ascoli. Problem 3.8.
Assume that an infinite-dimensional lcs E is Ascoli and contains no infinite-dimensional compact subsets. Is E topologically isomorphic to the space L ( ω ) ? Equiconvergence of topological spaces and proof of Theorem 1.2
In this section we establish two results related to equiconvergence in topological spaces.
Theorem 4.1.
If a topological space X admits an ω ω -base at a point x ∈ X , then X is ( ω , ω ) k -equiconvergent at the point x .Proof. Let ( U f ) f ∈ ω ω be an ω ω -base at x . To show that X is ( ω , ω ) k -equiconvergent at x ,fix an indexed family { x α } α ∈ ω ⊆ { s ∈ X ω : lim n →∞ s ( n ) = x } of sequences that converge to x . For every α ∈ ω consider the function µ α : ω ω → ω assigning to each f ∈ ω ω the smallest number n ∈ ω such that { x α ( m ) } m ≥ n ⊆ U f . It iseasy to see that the function µ α : ω ω → ω is monotone.For every n ∈ ω and finite function t ∈ ω n , let ω ωt = { f ∈ ω ω : f ↾ n = t } . By [4, Lemma2.3.5], for every f ∈ ω ω there exists n ∈ ω such that µ α [ ω ωf ↾ n ] is finite. Let T α be the set of ω -BASE AND INFINITE-DIMENSIONAL COMPACT SETS IN LOCALLY CONVEX SPACES 9 all finite functions t ∈ ω <ω = S n ∈ ω ω n such that µ α [ ω ωt ] is finite but for any τ ∈ ω <ω with τ ⊂ t the set µ α [ ω ωτ ] is infinite. It follows from [4, Lemma 2.3.5] that for every f ∈ ω ω there exists a unique t f ∈ T α such that t f ⊂ f .Let δ α ( f ) = max µ α [ ω ωt f ] ≥ µ α ( f ). It is clear that the function δ α : ω ω → ω is continuousand hence δ α is an element of the space C p ( ω ω , ω ) of continuous functions from ω ω to ω .Here we endow ω ω with the product topology. The function space C p ( ω ω , ω ) is endowedwith the topology of poitwise convergence. By Michael’s Proposition 10.4 in [19], the space C p ( ω ω , ω ) has a countable network.Consider the function δ : ω → C p ( ω ω , ω ) , δ : α δ α , and observe that δ α ( f ) ≥ µ α ( f )for any α ∈ ω and f ∈ ω ω .Since the space C p ( ω ω , ω ) has countable network, there exists a sequence { α n } n ∈ ω ⊆ ω of pairwise distinct ordinals such that the sequence ( δ α n ) n ∈ ω converges to δ α in thefunction space C p ( ω ω , ω ). We claim that the sequence ( x α n ) n ∈ ω witnesses that X is ( ω , ω ) k -equiconvergent at x . Given any open neighborhood O x ⊆ X of x , find f ∈ ω ω such that U f ⊆ O x . Since the sequence ( x α ( n )) n ∈ ω converges to x , there exists m ∈ ω such that { x α ( n ) } n ≥ m ⊆ U f . Since the sequence ( δ α n ) n ∈ ω converges to δ α in C p ( ω ω , ω ) we canreplace m by a larger number and additionally assume that δ α n ( f ) = δ α ( f ) for all n ≥ m .Choose a number l ≥ δ α ( f ) such that for every n < m and k ≥ l we have x α n ( k ) ∈ O x .On the other hand, for every n ≥ m and k ≥ l we have k ≥ l ≥ δ α ( f ) = δ α n ( f ) ≥ µ α n ( f )and hence x α n ( k ) ∈ U f ⊆ O x . (cid:3) Another property implying the ( ω , ω ) p -equiconvergence is the existence of a countable cs • -network. First we introduce the necessary definitions.Let x be a point of a topological space X . We say that a sequence { x n } n ∈ ω ⊆ X accumulates at x if for each neighborhood U ⊆ X of x the set { n ∈ ω : x n ∈ U } is infinite.A family N of subsets of X is defined to be • an s ∗ -network at x if for any neighborhood O x ⊆ X of x and any sequence { x n } n ∈ ω ⊆ X that accumulates at x there exists N ∈ N such that N ⊆ O x and the set { n ∈ ω : x n ∈ N } is infinite; • a cs ∗ -network at x ∈ X if for any neighborhood O x ⊆ X of x and any sequence { x n } n ∈ ω ⊆ X that converges to x there exists N ∈ N such that N ⊆ O x and theset { n ∈ ω : x n ∈ N } is infinite; • a cs • -network at x if for any neighborhood O x ⊆ X of x and any sequence { x n } n ∈ ω ⊆ X that converges to x there exists N ∈ N such that N ⊆ O x and N contains somepoint x n . • a network at x if for any neighborhood O x ⊆ X the union S { N ∈ N : N ⊆ O x } isa neighborhood of x ;It is clear that for any family N of subsets of a topological space X and any x ∈ X wehave the following implications.( N is an s ∗ -network at x ) (cid:11) (cid:19) ( N is a network at x ) (cid:11) (cid:19) ( N is a cs ∗ -network at x ) + ( N is a cs • -network at x ) Theorem 4.2.
If a topological space X has a countable cs • -network at a point x ∈ X , then X is ( ω , ω ) p -equiconvergent at x .Proof. Let N be a countable cs • -network at x and { x α } α ∈ ω ⊆ { s ∈ X ω : lim n →∞ s ( n ) = x } . Endow the ordinal ω + 1 = ω ∪ { ω } with the discrete topology. For every α ∈ ω considerthe function δ α : N → ω + 1 assigning to each N ∈ N the smallest number n ∈ ω suchthat x α ( n ) ∈ N if such number n exists, and ω if x n / ∈ N for all n ∈ ω . Since ( ω + 1) N isa metrizable separable space, the uncountable set { δ α } α ∈ ω ⊆ ( ω + 1) N contains a non-trivial convergent sequence. Consequently, we can find a sequence ( α n ) n ∈ ω of pairwise distinct countable ordinals such that the sequence ( δ α n ) n ∈ ω converges to δ α inthe Polish space ( ω + 1) N . We claim that the sequence ( x α n ) n ∈ ω witnesses that the space X is ( ω , ω ) p -equiconvergent. Fix any neighborhood U ⊆ X of zero.Since N is an cs • -network, there exists N ∈ N and n ∈ ω such that x n ∈ N ⊆ U . Hence d := δ α ( N ) ≤ n. Since the sequence ( δ α n ) n ∈ ω converges to δ α , there exists l ∈ ω such that δ α k ( N ) = δ α ( N ) = d for all k ≥ l . Then for every k ≥ l we have x α k ( d ) ∈ N ⊆ U . (cid:3) The following proposition (connecting ω ω -bases with networks) is a corollary of Theorem6.4.1 in [4]. Proposition 4.3. If ( U α ) α ∈ ω ω is an ω ω -base at a point x of a topological space X , then ( T β ∈↑ α U β ) α ∈ ω <ω is a countable s ∗ -network at x . Here ↑ α = { β ∈ ω ω : α ⊂ β } for any α ∈ ω <ω = S n ∈ ω ω n . As a consequence of the results presented above about the ( κ, λ ) p -equiconvergence andthe ( κ, λ )-tall bornology for a lcs E , we propose the following proof of Theorem 1.2. Proof of Theorem 1.2.
If a lcs E has an ω ω -base, then by Theorem 4.1, the space E is( ω , ω ) k -equiconvergent and hence ( ω , ω ) p -equiconvergent. The ( ω , ω ) p -equiconvergenceof E also follows from Proposition 4.3 and Theorem 4.2. Next, by Proposition 3.3, thespace E has ( ω , ω )-tall bornology, which means that each uncountable set in E containsan infinite bounded set. If E has an uncountable Hamel basis H , then H contains aninfinite bounded linearly independent set, and by Theorem 2.1 the space E contains aninfinite-dimensional compact set. (cid:3) Radial networks and another proof of Theorem 1.2
A family N of subsets of a linear topological space E is called a radial network if forevery neighborhood of zero U ⊆ E and every every x ∈ E there exist a set N ∈ N and anonzero real number ε such that ε · x ∈ N ⊆ U .The following theorem is a “linear” modification of Theorem 4.2. ω -BASE AND INFINITE-DIMENSIONAL COMPACT SETS IN LOCALLY CONVEX SPACES 11 Theorem 5.1.
If a lcs E has a countable radial network, then each uncountable subset in E contains an infinite bounded subset.Proof. Let N be a countable radial network in E , and let A be an uncountable set in E .Endow the ordinal ω + 1 = ω ∪ { ω } with the discrete topology.For every α ∈ A consider the function δ α : N → ω + 1 assigning to each N ∈ N theordinal δ α ( N ) = min { n ∈ ω + 1 : 2 − n · α ∈ [ − , · N } . Here we assume that 2 − ω = 0.Since ( ω + 1) N is a metrizable separable space, the uncountable set { δ α } α ∈ A ⊆ ( ω +1) N contains a non-trivial convergent sequence. Consequently, we can find a sequence { α n } n ∈ ω ⊆ A of pairwise distinct points of A such that the sequence ( δ α n ) n ∈ ω converges to δ α in the Polish space ( ω + 1) N .We claim that the set { α n } n ∈ ω is bounded in X . Fix any neighborhood U ⊆ X of zero.Since N is a radial network, there exist a set N ∈ N and a nonzero real number ε suchthat ε · α ∈ N ⊆ U . Then d := δ α ( N ) ∈ ω . Since the sequence ( δ α n ) n ∈ ω converges to δ α ,there exists l ∈ ω such that δ α k ( N ) = δ α ( N ) for all k ≥ l . Then for every k ≥ l we have2 − d · α k ∈ [ − , · N ⊆ [ − , · U and hence { α k } k ≥ l ⊆ [ − d , d ] · U , which implies that the family ( α n ) n ∈ ω is bounded in X . (cid:3) The implication (1) ⇒ (7) in the following theorem provides an alternative proof ofTheorem 1.2, announced in the introduction. Theorem 5.2.
For a lcs E consider the following properties: (1) E has an ω ω -base; (2) E has a countable s ∗ -network at zero; (3) E has a countable cs ∗ -network at zero; (4) E has a countable cs • -network at zero; (5) E has a countable radial network at zero; (6) each uncountable set in E contains an infinite bounded subset; (7) E contains an infinite-dimensional compact set.Then (1) ⇒ (2) ⇒ (3) ⇒ (4) ⇒ (5) ⇒ (6) . If E has uncountable Hamel basis, then (6) ⇒ (7) .Proof. The implication (1) ⇒ (2) follows from Proposition 4.3. The implications (2) ⇒ (3) ⇒ (4) are trivial and (4) ⇒ (5) follows from the observation that every cs • -networkat zero in the space E is a radial network for E . The implication (5) ⇒ (6) is proved byTheorem 5.1.If E has an uncountable Hamel basis H , then by (6), there exists an infinite bounded set B ⊆ H . By Theorem 2.1, the space E contains an infinite-dimensional compact set. (cid:3) Problem 5.3.
Is there an lcs E that has a countable radial network but does not have acountable cs • -network at zero? Applications to spaces C p ( X )A family { B α : α ∈ ω ω } of bounded (compact) sets covering a lcs E is called a bounded(compact) resolution if B α ⊆ B β for each α ≤ β . If additionally every bounded (compact)subset of E is contained in some B α , we call the family { B α : α ∈ ω ω } a fundamentalbounded (compact) resolution of E . Example 6.1.
Let E be a metrizable lcs with a decreasing countable base ( U n ) n ∈ ω ofabsolutely convex neighbourhoods of zero. For α = ( n k ) k ∈ ω ∈ ω ω put B α = T k ∈ ω n k U k andobserve that { B α : α ∈ ω ω } is a fundamental bounded resolution in E .A Tychonoff space X is called pseudocompact if each continuous real-valued function on X is bounded.The first part of the following (motivating) result has been proved in [18]; since this isnot published yet, we add a short proof. Proposition 6.2.
For a Tychonoff space X the following assertions are equivalent: (1) The space C k ( X ) is covered by a sequence of bounded sets. (2) The space C p ( X ) is covered by a sequence of bounded sets. (3) X is pseudocompact.Moreover, the following assertions are equivalent: (4) C p ( X ) is covered by a sequence of bounded sets but is not covered by a sequence offunctionally bounded sets. (5) X is pseudocompact and contains a countable subset which is not closed in X ornot C ∗ -embedded in X .Proof. (1) ⇒ (2) is clear. (2) ⇒ (3): Assume C p ( X ) is covered by a sequence of boundedsets but X is not psudocompact. Then C p ( X ) contains a complemented copy of R ω , see [1].But R ω cannot be covered by a sequence of bounded sets, otherwise would be σ -compact.(3) ⇒ (1): If X is pseudocompact, then for every n ∈ N the set B n = { f ∈ C ( X ) :sup x ∈ X | f ( x ) | ≤ n } is bounded in C k ( X ) and S n ∈ N B n = C k ( X ).The equivalence (4) ⇔ (5) follows from [24, Problem 399]: C p ( X ) is covered by asequence of functionally bounded subsets o C p ( X ) if and only if X is pseudocompact andevery countable subset of X is closed and C ∗ -embedded in X . (cid:3) Example 6.3. C p ([0 , ω )) is covered by a sequence of bounded sets but is not covered bya sequence of functionally bounded sets.By [10], C p ( X ) has a bounded resolution if and only if there exists a K -analytic space L such that C p ( X ) ⊆ L ⊆ R X . The problem when C p ( X ) has a fundamental boundedresolution is easier. As a simple application of Theorem 1.2 we prove the following Proposition 6.4.
For a Tychonoff space X consider the following assertions: (1) C p ( X ) admits a fundamental bounded resolution { B α : α ∈ ω ω } . (2) X is countable. (3) R X = S α ∈ ω ω B α R X for a fundamental bounded resolution { B α : α ∈ ω ω } in C p ( X ) . ω -BASE AND INFINITE-DIMENSIONAL COMPACT SETS IN LOCALLY CONVEX SPACES 13 (4) The strong (topological) dual L β ( X ) of C p ( X ) is a cosmic space, i.e. a continuousimage of a metrizable separable space. (5) C p ( X ) is a large subspace of R X , i.e. for every mapping f ∈ R X there is a boundedset B ⊆ C p ( X ) such that f ∈ B R X .Then (1) ⇔ (2) ⇔ (3) ⇔ (4) ⇒ (5) but (5) ⇒ (2) fails even for compact spaces X . The implication (1) ⇒ (2) was recently proved by Ferrando, Gabriyelyan and K¸akol [9](with the help of cs ∗ -networks). We will derive this implication from Theorem 1.2. Proof. (1) ⇒ (2): If C p ( X ) has a fundamental bounded resolution { B α : α ∈ ω ω } , thenthe sets U α = { ξ ∈ L β ( X ) : sup f ∈ B α | ξ ( f ) | ≤ } form an ω ω -base in L β ( X ). By [14],every bounded set in L β ( X ) is finite-dimensional. Applying Theorem 1.2, we concludethat the Hamel basis X of the lcs L β ( X ) is countable. (2) ⇒ (1) is clear. (2) ⇒ (3) ∧ (5):Since C p ( X ) is dense in the metrizable space R X , the claims hold. (2) ⇒ (4): If X iscountable, then L β ( X ) has a fundamental sequence of compact sets covering L β ( X ) and[19, Proposition 7.7] implies that L β ( X ) is an ℵ -space, hence cosmic. (4) ⇒ (2): If L β ( X )is cosmic, then it is separable, and [12, Corollary 2.5] shows that X is countable. (5) ; (2): C p ( X ) over every Eberlein scattered, compact X satisfies (5), see [13]. (cid:3) Item (5) in Proposition 6.4 is strictly connected with the following result.
Theorem 6.5. ([13], [12])
For a Tychonoff space X , the following conditions are equivalent :(i) C p ( X ) is distinguished, i.e. the strong dual L β ( X ) of the space C p ( X ) is bornolog-ical. (ii) The strong dual L β ( X ) of the space C p ( X ) is a Montel space. (iii) C p ( X ) is a large subspace of R X . (iv) The strong dual L β ( X ) of the space C p ( X ) carries the finest locally convex topology. The following is a linear counterpart to item (4) in Proposition 6.4.
Remark 6.6.
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