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We prove that every homogeneous countable dense homogeneous topological space containing a copy of the Cantor set is a Baire space. In particular, every countable dense homogeneous topological vector space is a Baire space. It follows that, for any nondiscrete metrizable space $X$, the function space $C_p(X)$ is not countable dense homogeneous. This answers a question posed recently by R. Hern\'andez-Guti\'errez. We also conclude that, for any infinite dimensional Banach space $E$ (dual Banach space $E^\ast$), the space $E$ equipped with the weak topology ($E^\ast$ with the weak$^\ast$ topology) is not countable dense homogeneous. We generalize some results of Hru\v{s}\'ak, Zamora Avil\'es, and Hern\'andez-Guti\'errez concerning countable dense homogeneous products., Comment: slightly modified and expanded version

This paper deals with the existence of periodic solutions and attractiveness for some partial functional differential equations in Banach spaces. We assume that the first linear part generates a strongly continuous semigroup, while the delayed part is periodic with respect to the first argument. We prove that the existence of a bounded solution implies the existence of a periodic solution. Several results regarding uniqueness and global attractiveness of periodic solutions are also established. The analysis relies on a fixed point theorem of Chow and Hale’s type and uses some arguments of weak topology. Our theorems extend in a broad sense some new and classical related results. An application to a transport equation with delay is also presented.

In this paper we are concerned with the study of a long-standing open problem posed by Lau in 1976. This problem is about whether the left amenability property of the space of left uniformly continuous functions of a semitopological semigroup is equivalent to the existence of a common fixed point for every jointly weak* continuous norm nonexpansive action on a nonempty weak* compact convex subset of a dual Banach space. We establish in this paper a positive answer.

The aim of this note is study the topology generated by Lipschitz slices in the unit sphere of a Banach space. We prove that the above topology agrees with the weak topology in the unit sphere and, as a consequence, we obtain Lipschitz characterizations of classical linear topics in Banach spaces, as Radon-Nikodym property, convex point of continuity property and strong regularity, which shows that the above classical linear properties only depend on the natural uniformity in the Banach space given by the metric and the linear structure., Comment: 11 pages

In 1979, Trotman proved, using the techniques of the Thom transversality theorem, that under some conditions on the dimensions of the manifolds under consideration, openness of the set of maps transverse to a stratification in the strong (Whitney) topology implies that the stratification is ( a ) (a) -regular. Here we first discuss the Thom transversality theorem for the weak topology and then give a similiar kind of result for the weak topology under very weak hypotheses. Recently, several transversality theorems have been proved for complex manifolds and holomorphic maps. In view of these transversality theorems we also prove a result analogous to Trotman’s result in the complex case.

We show that the unit ball of l p (Γ) in its weak topology is a continuous image of σ 1 (Γ) N , and we deduce some combinatorial properties of its lattice of open sets which are not shared by the balls of other equivalent norms when r is uncountable.

In this paper, we obtain an explicit formula for the interior of the domain of a maximal monotone multifunction in terms of its Fitzpatrick function.

We show that ℓ ∞ / c 0 = C ( N ∗ ) \ell _\infty /c_0=C(\mathbb {N}^*) with the weak topology is not an intersection of ℵ 1 \aleph _1 Borel sets in its Čech-Stone extension (and hence in any compactification). Assuming (CH), this implies that ( C ( N ∗ ) , w e a k ) (C(\mathbb {N}^*),\mathrm {weak}) has no continuous injection onto a Borel set in a compact space, or onto a Lindelöf space. Under (CH), this answers a question of Arhangel’skiĭ.

In this paper we extend classical results concerning generalized convolution structures on measure spaces. Given a locally compact Hausdorff space X, we show that a compactly bounded convolution of point masses that is continuous in the topology of weak convergence with respect to C c (X) can be extended to a general convolution of measures which is separately continuous in the topology of weak convergence with respect to C b (X).

It is shown that the relaxation of the integral functional involving argument deviations I(u):= ∫ Ω f(x,(u i (g ij (x))} k,t i,j=1 ) dμΩ(x), in weak topology of a Lebesgue space (L p (Θ, μ Θ )) k (where (Ω, Σ(Ω),μ Ω ) and (Θ,Σ(Θ),μ μ Θ ) are standard measure spaces, the latter with nonatomic measure), coincides with its convexification whenever the matrix of measurable functions g ij : Ω → Θ satisfies the special condition, called unifiability, which can be regarded as collective nonergodicity or commensurability property, and is automatically satisfied only if k = l = 1. If, however, either k > 1 or l > 1, then it is shown that as opposed to the classical case without argument deviations, for nonunifiable function matrix {g ij } one can always construct an integrand f so that the functional I itself is already weakly lower semicontinuous but not convex.

We introduce and study the notion of pseudo-uniform convergence which is a weaker variant of quasi-uniform convergence. Applications include the following nonstandard characterization of weak convergence. Let X X be an infinite set, B ( X ) B(X) the Banach space of all bounded real-valued functions on X , X, { f n : n ∈ N } \{f_{n}: n\in N\} a bounded sequence in B ( X ) , B(X), and f ∈ B ( X ) . f\in B(X). Then the sequence converges weakly to f f if and only if the convergence is pointwise on X X and, for each strictly increasing function σ : N → N \sigma :N\to N , each x ∈ ∗ X x\in ^{*}X , and each n ∈ ∗ N ∞ n\in ^{*}N_{\infty } , there is an unlimited m ≤ n m\leq n such that ∗ f ∗ σ ( m ) ( x ) ≃ ∗ f ( x ) ^{*}f_{ ^{*}\sigma (m)}(x) \simeq ^{*}f(x) .

Let the space Id(A) of closed two-sided ideals of a Banach algebra A carry the weak topology. We consider the following property called normality (of the family of finite subsets of A): if the net (Ii)i in Id(A) converges weakly to I, then for all a E A\I we have liminfi Ila + I ll > 0 (e.g. C*-algebras, L1 (G) with compact G,... ). For a commutative Banach algebra normality is implied by spectral synthesis of all closed subsets of the Gelfand space A(A), the converse does not always hold, but it does under the following additional geometrical assumption: inf{Ipll 0211;P1,(P2 E A(A),pi $o : } > 0.

For a locally convex Hausdorff topological vector space E E and for a system V V of weights vanishing at infinity on a locally compact Hausdorff space X X , let C V 0 ( X , E ) CV_0(X,E) be the weighted space of E E -valued continuous functions on X X with the locally convex topology derived from the seminorms which are weighted analogues of the supremum norm. A characterization of the orthogonality preserving (Lamperti-type) operators on C V 0 ( X , E ) CV_0(X,E) is presented in this paper.

The distributions of two–block–factors (f(ηi, ηi+1); i ≥ 1) arising from i.i.d. sequences (ηi; i ≥ 1) are observed to coincide with the distributions of the superdiagonals (ζi,i+1; i ≥ 1) of jointly exchangeable and dissociated arrays (ζi,j ; i, j ≥ 1). An inequality for superdiagonal probabilities of the arrays is presented. It provides, together with the observation, a simple proof of the fact that a special one–dependent Markov sequence of Aaronson, Gilat and Keane (1992) is not a two–block factor.

Given separable Frechet spaces, E, F, and G, let L ( E , F ) , L ( F , G ) \mathcal {L}(E,F),\mathcal {L}(F,G) , and L ( E , G ) \mathcal {L}(E,G) denote the space of continuous linear operators from E to F , F to G, and E to G, respectively. We topologize these spaces of operators by any one of a family of topologies including the topology of pointwise convergence and the topology of compact convergence. We will show that if ( X , F ) (X,\mathcal {F}) is any measurable space and both A : X → L ( E , F ) A:X \to \mathcal {L}(E,F) and B : X → L ( F , G ) B:X \to \mathcal {L}(F,G) are Borelian, then the operator composition B A : X → L ( E , G ) BA:X \to \mathcal {L}(E,G) is also Borelian. Further, we will give several consequences of this result.

In this paper we deal with the (o)-convergence and the order topology in the hilbertian logic L ( H ) \mathcal {L}(H) of closed subspaces of a separable Hilbert space H. We compare the order topology on L ( H ) \mathcal {L}(H) with some other topologies. The main result is a theorem which asserts that the weak convergence of a sequence of unit vectors to zero in H is equivalent to the convergence of the sequence of one-dimensional subspaces generated by these vectors to the zero subspace in the order topology on L ( H ) \mathcal {L}(H) .

Using a technique developed by Louveau and Saint Raymond, we find the complexity of the space of probability measures in the Borel hierarchy: if X X is any non-Polish Borel subspace of a Polish space, then P ( X ) P(X) , the space of probability Borel measures on X X with the weak topology, is always true Π ξ 0 {\boldsymbol {\Pi }^{\boldsymbol {0}}_{\boldsymbol {\xi }}} , where ξ \xi is the least ordinal such that X X is Π ξ 0 {\boldsymbol {\Pi }^{\boldsymbol {0}}_{\boldsymbol {\xi }}} .

In this paper, we obtain necessary and sufficient conditions for a c 0 {c_0} -semigroup with continuity in the uniform operator topology for t > 0 t > 0 in Hilbert space in terms of the spectral property of its infinitesimal generator. Then we get the characteristic theorem for the compact semigroup.

Let ⟨ X , d ⟩ \left \langle {X,d} \right \rangle be a complete and separable metric space. The Wijsman topology on the nonempty closed subset CL ⁡ ( X ) \operatorname {CL}\left ( X \right ) of X X is the weakest topology on CL ⁡ ( X ) \operatorname {CL}\left ( X \right ) such that for each x x in X X , the distance functional A → d ( x , A ) A \to d\left ( {x,A} \right ) is continuous on CL ⁡ ( X ) \operatorname {CL}\left ( X \right ) . We show that this topology is Polish, and that the traditional extension of the topology to include the empty set among the closed sets is also Polish. We also compare the Borel class of a closed valued multifunction with its Borel class when viewed as a single-valued function into CL ⁡ ( X ) \operatorname {CL}\left ( X \right ) , equipped with Wijsman topology.

We give an alternate description of algebras in the class of ultraweakly closed subspaces of q(X) via the preannihilator. We then apply this result to show that proper ultraweakly closed algebras of bounded operators on an infinite-dimensional Hilbert space X have infinite codimension. We also use this alternate description of algebras to say something the structure of rank-one operators in unicellular algebras. We begin with some basic definitions and notation. For ', an infinitedimensional Hilbert space, let 5F(X) denote the set of all bounded linear operators on X and let Y(X) denote the set of all trace-class operators on X'. Then Y('), equipped with the trace norm, is a Banach space whose dual is 5( ). The duality is given by the linear functional on 7(X) x (X) defined by (t, x) tr(tx) for t EzY(X), x E (X), where tr denotes the trace. Thus we get a w*-topology on 5(X), which is also known as the ultraweak topology. The weak-operator topology (which we shall refer to as the weak topology) is actually weaker than the ultraweak topology, so all the results stated in this paper for ultraweakly closed subspaces are true for weakly closed subspaces. An excellent exposition of the role of this duality theory in invariant subspace theory is [ 1]. We shall follow the notation of [ 1], which the reader can consult for more background. As usual, we can use the above duality to define preannihilators. For ? an ultraweakly closed subspace of (X), the preannihilator is X1 = {t EY7(X) I tr(tm) = 0 for all m Ecz}. The codimension of A'(codim(.')) is the vector space dimension of {f(X)/ X'}. If we identify all infinite cardinals, then this is also equal to the vector space dimension of X1 . Given x, y E X, the operator x 0 y is the rank-one operator defined by x X y(z) = (z, y)x for z E X. Received by the editors July 3, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 47D25. ? 1990 American Mathematical Society 0002-9939/90 $1.00 + $.25 per page

We give an elementary proof of the principle of local reflexivity. We use only elementary functional analysis to give a simple and short proof of the version of the "principle of local reflexivity" proved in [2], which is an improvement of the original version given in [3]. Other short proofs can be found in [1] and [4]. We use standard notation for Banach spaces. By X, X* and X**, we denote a real or complex Banach space, its first dual and its second dual respectively; we identify X with the canonical copy of X contained in X**; given a Banach space Y, we write By := y : IIyll 0, an e-isometry T: E -> Y is an operator for which 1 e Y be an operator, z E intBx** and y E Y such that IIT**z yll 0 there exists an e-isometry T : E -> X such that T |Enx= id IEnx, and f(Te) = e(f) for all f C F and all e E E. Proof. Let dimE = n and dimE n X = n k. Let (yj, hj))1 be a biorthogonal system in E x E* such that Ilyjll = 1 e and span{yj}^L l = E n X. The identity id : E -X** can be given as id(e) = Ej1 hj(e)yj. We shall find v1,... , Vk in X so that the operator T: E -> X defined by T(e) = Z>=1 hj (e)vj + Zj=k+l hj(e)yj is an e-isometry. Hence, the condition T iEnx= id IEnX will be satisfied automatically. Let W := Xk endowed with the norm ||(xj)=ill| = supj |lxjll, and select 0 < a < min{2/5, (1 )-1 1, 6(E=L1 hjll-)-1}. Fix {ff}Jj1 a basis in F, {ej}l'L1 Received by the editors September 27, 1996 and, in revised form, August 18, 1997. 1991 Mathematics Subject Classification. Primary 46B20, 46B10.

Let E be a Hausdorff topological vector space having a Schauder basis {bi} and coordinate functionals {fi}. Let ar(E, F) be the weak topology on E induced by F = {fi: i E N}. We show that if a series in E is subseries convergent with respect to a(E, F), then it is subseries convergent with respect to the original topology of E. In [S] Stiles established what seems to be the first version of the Orlicz-Pettis Theorem for nonlocal spaces. Indeed, Kalto,n remarks in [K1] that Stiles's result motivated his far-reaching generalizations of the Orlicz-Pettis Theorem for series in a topological group [K2]. Stiles's version of the Orlicz-Pettis Theorem is for F-spaces with a Schauder basis, and his proof uses the metric properties of the space. Other proofs of Stiles's result have been given in [B] and [Swl], and both of these proofs also rely heavily on the metric. Kalton has also given a substantial generalization of Stiles's result in the Corollary to Theorem 3 in [K2]; this generalization is also for metric linear spaces. In this note we will give a proof of Stiles's Orlicz-Pettis Theorem for arbitrary Hausdorff topological vector spaces which have a Schauder basis. Our proof uses a very interesting result of Antosik on the convergence of double series [A]. Throughout this note let E be a Hausdorff topological vector space with a Schauder basis {bi} and coordinate functionals {f } . Let F = {f: i E N} , and let a(E, F) be the weak topology on E from the duality between E and F. If z is any vector topology on E, a series E xj in E is said to be z subseries convergent if for every subsequence {xnj} of {x;} the subseries I x,j is T convergent in E. We show that every series I: xj in E which is a(E, F) subseries convergent is subseries convergent in the original topology of E. For the proof of our result we use the following theorem of Antosik on iterated double series [A]. Theorem 1 (Antosik). Let xij E E for i, j E N. Iffor every increasing sequence of positive integers {nj} the iterated series 1:00 Zi 10 xin1 converges, then the double series >, j xij converges. In particular, the iterated series E' 1 1i Xij converges and equals ' I E'I xij . Received by the editors February 2, 1993 and, in revised form, May 7, 1993. 1991 Mathematics Subject Classification. Primary 46A35. ? 1994 American Mathematical Society 0002-9939/94 $1.00 + $.25 per page

Let X denote the group of all automorphisms of a finite Lebesgue space equipped with the weak topology. For T E A, let UrT denote its maximal spectral type. Theorem 1. There is a dense G3 subset G c X such that, for each T E G and k(1), . k(l) E Z+, k'(1), ... , k'(l') E Z+, the convolutions UTk(1) * * UTk(U) and a TkI(1) *... * UTkO(1) are mutually singular, provided that (k(1) . k(l)) is not a rearrangement of (k'(1), ... , k'(l')). Theorem 1 has the following consequence. Theorem 2. X has a dense G3 subset F c G such that for T E F the following holds: For any k: N -. Z {O} and 1 E Z {O}, the only way that TI, or any factor thereof can sit as a factor in T k(l) x T k(2) X *.* is inside the ith coordinate c-algebra for some i with k(i) = 1 . Theorem 2 has applications to the construction of certain counterexamples, in particular nondisjoint automorphisms having no common factors and weakly isomorphic automorphisms that are not isomorphic.

Talagrand [ 10] gives an example of a Banach space with weak topology which is not a Radon space, independently of their weight. This result gives an answer to a question formulated by Schwartz [9]. In this paper, following the papers of Drewnowski and Roberts [1] and Talagrand [10], we prove that the classical space (lo /c0 , weak) is not a Radon space. Introduction. A Hausdorff topological space E is said to be a Radon space if every finite Borel measure is a Radon measure; i.e., (A) = {8(K): K c A, K compact} for each Borel subset A of E. We shall say that a cardinal a is of measure zero (resp. nonmeasurable) if there is not a real-valued, diffuse, nontrivial measure (resp. {0, 1 }-valued), on the power set of a set A with cardinal a. The weight (density character) of a topological space E is the smallest cardinal such that there exists in E a dense subset A with this cardinal. A topological space E has the a-property of Lindelof, a a transfinite cardinal, if for each family (Gi)icE of open subsets of E there exists J c I such that card(J) < a and Uic Gi = Uij Gi* The smallest cardinal a such that E has the a-property of Lindelof is called the L-weight of E. Likewise, E is a Flock space if for every well-ordered family (Gi)LEI of open sets, with Ha = Gc\ Ufl

The group of measure preserving transformations of the unit interval equipped with the weak topology is an absolute retract, hence is homeomorphic to a separable Hilbert space.

This note deals with the problem of minimizing a real-valued function f on a weakly closed subset of a reflexive Banach space. We use a mild monotonicity assumption introduced by P. Hess (11 ] on the derivative f' of f to get the weak lower semicontinuity of f. We show that one can dispense with any continuity assumption on f', so that we get a true generalization of F. E. Browder's results [4]. The relevance of the monotonicity property to the calculus of variations is shown by an example. 0. Introduction. In [4] F. E. Browder brought forward the role of monotonicity assumptions on the derivative f' of a function f: X -> R, X a reflexive Banach space, to ensure its weak lower semicontinuity. P. Hess showed [1 1] that the monotonicity assumption of f' can be relaxed providedf' is continuous. In this note we prove that this restriction is unnecessary, so that F. E. Browder's result is encompassed within the more general frame of [1 1]. 1. Preliminaries. Let (X, T) be a locally convex topological vector space (l.c.s.)with dual X'. We denote by a the weak topology a(X, X') on X and by a, the sequential topology associated with a: a subset A of X is a,-closed if and only if A contains the limit of any a-convergent sequence of A. It is easily seen that a and a, have the same convergent sequences which are called weakly convergent sequences. The lower semicontinuity (l.s.c.) of a map f: X -> R with respect to ac amounts to its sequential weak lower semicontinuity. The following definition is an extension to multivalued mappings of a definition of P. Hess [11]. DEFINITION 1. A multivalued mapping F: X -> X' is called pertinent if it satisfies: If (xv) is a sequence of dom F weakly converging to x E (P) dom F, if (x,) is a sequence of X' with xn E F(xn) for every n, then lim sup > 0. As observed by P. Hess, condition (P) is an extremely mild monotonicity assumption. Multivalued monotone mappings, multivalued pseudo-monotone Received by the editors March 26, 1976 and, in revised form, January 5, 1977. AMS (MOS) subject classifications (1970). Primary 47H05, 49A25, 49A50.

If A A is an S W SW algebra of real-valued functions on a set X X equipped with the weak topology for A A , and if A A separates its zero sets, then many results valued for C b ( X ) {C^b}(X) equipped with a strict topology remain true when A A is equipped with a strict topology. The concepts of α \alpha -additivity and tight positive linear operators are introduced. It is shown that if T T is a positive linear map on A A into z z -separating S W SW algebra B B and if T ( 1 A ) = 1 B T({1_A}) = {1_B} , then there exists a continuous function ϕ \phi on Y Y (the domain of elements in B B ) into X X such that T f ( y ) = f ( ϕ ( y ) ) Tf(y) = f(\phi (y)) if and only if T T is an algebraic homomorphism and τ \tau -additive.

Let C C be a bounded closed convex subset of a strictly convex Banach space and let S S be a semigroup of nonexpansive self-mappings of C C which is convex and compact in the topology of weak point-wise convergence. If S S has the property that co ¯ R ( s 1 ) ∩ co ¯ R ( s 2 ) ≠ ∅ \overline {\operatorname {co} \,} \mathcal {R}({s_1}) \cap \overline {\operatorname {co} \,} \mathcal {R}({s_2}\;) \ne \emptyset whenever s 1 , s 2 ϵ S {s_1},\;{s_2}\epsilon S , then S S has a common fixed point and F ( S ) F(S) is a nonexpansive retract of C C .

Necessary and sufficient conditions for a uniform measure to have a tight extension are presented. From these results, conditions for the factors of a tight measure to be also tight are derived. 1. Introduction. In this paper we derive conditions for a uniform measure p on a uniform space (9C, %) to have an extension which is tight on (%, *5 ), where Iis a topology which is finer than the uniform topology of %. and where (%,§) is Hausdorff. These conditions are expressed in terms of the images of p in certain complete metric spaces associated with (9C, %). One consequence is that conditions for the factors of a tight measure to be also tight can be obtained. In §2 relevant properties of uniform measures are given, and a necessary and sufficient condition for extending a uniform measure as a tight measure is obtained. In §3 we consider the following question. Let £ be a topological vector space. Let p be a tight measure and vx,v2be uniform measures on E such that p = vx * v2. When are vx and v2 tight? Finally, in §4, we consider the case where (%

A uniform topology τ M {\tau _M} induced by a set M M of finite measures on a quantum logic L L is studied. If m m is a valuation on L L , the topology τ m {\tau _m} induced by { m } \{ m\} is equivalent to the topology induced by the pseudometric ρ ( a , b ) = m ( a Δ b ) \rho (a,b) = m(a\Delta b) . If the set M M of measures is large enough, the topology τ M {\tau _M} reflects in some sense the structure of L L : if L L is a continuous geometry and the measures are totally additive, τ M {\tau _M} is weaker than the order topology τ o {\tau _o} on L L . If L L is atomic, τ M {\tau _M} is stronger than τ o {\tau _o} . On a separable Hilbert space logic, τ M {\tau _M} coincides with the discrete topology. Some cases are found in which τ M = τ o {\tau _M} = {\tau _o} .

This paper characterizes in terms of weak topologies those bounded linear functionals on a subspace which have unique Hahn-Banach extensions to the whole linear normed space. The relationship to the Choquet boundary is discussed, and a Korovkin type theorem is obtained. We begin with a different proof of Korovkin's theorem in order to motivate that which follows. Theorem 1 (Korovkin [21). Suppose IL } is a sequence of positive lin ear functionals on C[O, 1] satisfying L (1) 1, L (I) -* c, and Ln(12) c2. (Here I is the identity function.) Then L (f) f(c) for all f 6 C[0, 1]. n Proof of Theorem 1. Identify C* with the regular Borel measures on 10, 1]. Identify L with a regular positive measure by L (f) = ffd1 n. Any subnet pa of }un has a further subnet ye converging w(C*, C) to some , and it follows that [ty(1) ,q'(l) = 111 In particular p? 1) = 1, (1) = c, and p(I2 = c2. Then p must be (, the point evaluation measure, for suppose A(jcC) K 1; then there is an open interval J containing c with tt(j) O. Then ? 0 (I_ c) 2 dtt> ,(I _c) 2dtt> rll(j)=r( (p) > O, a contradiction. Since every subnet of '.Un has a further subnet converging w(C*, C) to (c Sthen ln + ec in w(C*, C), hence Lnf = ff dn -ff dfcj= f(c) for all f 6 C[0, 1]. While this proof is admittedly less elementary than Korovkin's, it makes it clear that the heart of the matter is the fact that the point evaluation funcReceived by the editors October 9, 1973. AMS (MOS) subject classifications (1970). Primary46-00,46B99; Secondary 46J 20.

In this work we show that if X ∗ {X^ * } contains bounded sequences without weak* convergent convex blocks, then it contains an isometric copy of L 1 ( { 0 , 1 } ω 1 ) {L_1}\left ( {{{\left \{ {0,1} \right \}}^{{\omega _1}}}} \right ) .

This paper concerns the question: given an action of a topological group on a manifold and a point which is stationary for this action, what conditions will insure that the point is stable? It is possible that a good general answer to this question would include conditions on the global topology of the manifold. This paper, however, presents conditions on just the linear part of the action at the stationary point which are sufficient for its stability, and which in fact are sufficient for the stability of the stationary set near that point. Let G be a topological group and M a finite-dimensional C' manifold. An action of G on M is a continuous homomorphism a: G -* Diff'(M). Here Diff'(M), the group of C1 diffeomorphisms of M, is understood to carry the weak topology, in which convergence of a net of diffeomorphisms means uniform convergence on compact sets of the corresponding homeomorphisms of TM. The set of all actions of G on M itself has a topology, namely the compact-open topology which it inherits as a subset of the set of continuous maps from G to Diff'(M). A point of M is stationary for a if it is fixed by a(g) for every g E G. Finally, a stationary point p is stable if, given any neighborhood U of p, there is a neighborhood N of a such that each /3 Ee N has a stationary point in U. An action with stationary point p induces a linear action (that is, an action for which each element of G acts by a linear map) on the tangent space Mp, given by g " Da(g)p. This linear action is the focus of this paper, and I begin by looking at a general linear action y: G GL(E), where E is a finite-dimensional vector space. For u: G -*E, a map yu: G Aff(E) is defined by yu(g)x = y(g)x + u(g). yu will be an action exactly when (1) u is continuous and (2) u(gh) = y(g)u(h) + u(g) for all g, h E G. A pointy E E will be stationary for yu exactly when (3) u(g) = y y(g)y for all g E G. Maps satisfying (1) and (2) ("crossed homomorphisms") form a vector space; the

Given an algebra E and a total subspace E ′ E’ of its algebraic dual, we obtain necessary and sufficient conditions in terms of E ′ E’ for the existence of an A-convex or a locally m-convex topology on E compatible with duality ( E , E ′ ) (E,E’) . It has also been proved that if E with the weak topology w ( E , E ′ ) w(E,E’) is the closed linear hull of a bounded set and has hypocontinuous multiplication then it is locally m-convex.

It is proved that a necessary and sufficient condition for the group topology of any maximally almost periodic Hausdorff group to coincide with the weak topology of the group in which every complex-valued continuous almost periodic function on the group is continuous is that the group has the equivalent left and right uniform structures. The sufficiency of this condition generalizes the recent results of Glicksberg and Venkataraman concerning the group topology and the weak topology of an abelian Hausdorff group induced by the set of all continuous characters of the group.

Let H be a topological group contained in a group G. A topology which makes G a topological group inducing the given topology on H is called an extending topology. The set of all extending topologies forms a complete semilattice in the lattice of group topologies on G. The structure of this semilattice is studied by considering normal subgroups which intersect H in the identity. Let G be a group and ? the collection of continuous topologies on G. If { T }a E r is any collection of topologies in ?, we can create a new topological group C=H Iler(G, T.). The embedding of G into C along the diagonal is an algebraic embedding. The relative topology on G in C is the supremum topology on G relative to {T,}a er. Since G is a topological group we see that the subgroup G is also a topological group when given this topology. Hence ? is closed under the operation Ta V T where T. V T,B denotes the supremum topology Ta and T,B Now suppose T. and T,B are in ? and 2 c ? is the collection of all topologies Ty in Y that satisfy Ty c T1 and Ty c T. a # 0 since the indiscrete topology on G is in ?Y. We define Ta A T = (Vy TY). Certainly Y is closed under A and hence (Y, V, A) forms a complete lattice. It should be noted that this lattice is different from the usual lattice of topologies on a set X since the intersection of two group topologies may not be a group topology. Let H be a topological group contained in a group G. A topology which makes G a topological group inducing the given topology on H is called an extending topology. The set of all extending topologies from H to G, o, is a complete subsemilattice of the complete lattice of group topologies on G. The purpose of this paper is to further study 46. DEFINITION. A group topology for H is said to be translatable if and only if for every neighborhood U of e in H and every g E G, the set gUg-' contains a neighborhood of e in H. If t is a translatable topology on H we define the translation topology TH* to be that topology on G which has { gUIg E U and U E t } as a basis. One way to create a topology on G is to find a homomorphism from G to a topological group G'. The weak topology on G relative to this homomorphism will make G into a topological group also. Occasionally this topology will be an extending topology. Let H be a subgroup of G endowed with the translatable Received by the editors May 16, 1985 and, in revised form, June 18, 1985. 1980 Mathematics Subject Classification. Primary 22A99. Kev words and phrases. Lattice, group topologies. ?1986 American Mathematical Society 0002-9939/86 $1.00 + $.25 per page

Corresponding to each family F of real-valued functions on a set X, there is a weakest topology on X for which F is equicontinuous. This equiweak topology is pseudometrizable and provides a characterization of metrizable topologies in terms of point-separating families of real-valued functions. Introduction. Let X be a set and let F be a collection of real-valued functions defined on X. In this paper, we show there always exists a weakest topology on X for which F is an equicontinuous family. We define this to be the equiweak topology induced by F and show that such topologies play a role in metrization theory analogous to the role of weak topologies in the theory of completely regular Hausdorff spaces. In particular, we show that a topology is metrizable if and only if it is an equiweak topology induced by a point-separating family of real-valued functions. We first establish some notation and terminology. All functions are defined on X and are real-valued. The collection of all such functions is denoted by RX. The weak topology induced on X by F c Rx is denoted by SF. The collection F separates points in X if for distinct points x1, x2 E X, there is some f E F such that f(xI) #f(x2). If X is equipped with a topology, C(X) will denote its collection of continuous functions. A collection F c Rx is equicontinuous at xo E X if, given E > 0, there is a neighborhood U of x0 so that If(x) f(xo)I < E for all x E U and all f E F. If F is equicontinuous at x0 for each x0 E X, then F is simply said to be equicontinuous. The equiweak topology. LEMMA 1. Let F c RX be equicontinuous at xo E X. Then for each p E X, the function X SUPF{ If(X) f(P)I A 1) is continuous at xo. PROOF. Fixp E X. Then for any x E X, we have that ISUPF{If(X) f(P)I A 1) SUpF{If(XO) f(P)I A Ill < SUPF({[If(x) -f(P)I A ] -[If(xo) -f(P)I A I1]1 < SUpF| I|f(X) -f(P)I If(xo) -f(P)I I) < SUPFF{If(X) -f(XO)}. Received by the editors November 17, 1978 and, in revised form, November 12, 1979. AMS (MOS) subject classifications (1970). Primary 54C30, 54E35.

A theorem on the coincidence of a mixed topology and the Mackey topology is given. The techniques of partitions of unity are used. As a corollary, a result of LeCam and Conway on the strict topology is obtained. Similar methods are applied to obtain a theorem of Collins and Dorroh. Introduction. In recent years the space C(S) of bounded, continuous complex-valued functions on a locally compact space S with the strict topology has received a great deal of attention. In [5] we have shown that this topology was a special case of a mixed topology in the sense of the Polish school. Many of the results on the strict topology are, in fact, special cases of general results on mixed topologies and, when this is the case, it seems to us to be desirable to present this approach since it displays the strict topology as part of a larger scheme rather than as an isolated phenomenon. In this note, we give an example of this process by considering the problem: When is the strict topology on C(S) the Mackey topology? LeCam and Conway have shown that this is the case when S is paracompact. In fact, there are a number of results of this type, that is, identifying mixed topologies with the Mackey topology (see Cooper [6], Stroyan 113]) and this suggests that there might be a general theorem of this type in the context of mixed topologies. In this note, we give such a theorem (11 below). Although the result on C(S) that we prove (12 below) is not new, it seemed to us worthwhile to give a proof of this type since it shows that the problem is mainly one in functional analysis and that measure theoretical techniques, for example, are unnecessary. In addition, we feel that the methods that we use are of some interest in themselves. We mention three points. It is known that the duality between sums and products in the category of locally convex spaces breaks down in the category of Banach spaces. We show how this duality can be recovered by using mixed topologies (5(i) and (ii)). The result 5(iii) can be regarded as an infinite dimensional version of Schur's theorem (if each E. is one dimensional we obtain the classical Schur theorem-of course, we have used the latter in the proof of 5(iii)). Thirdly, we use the technique of partitions of unity which were introduced by De Wilde [16] for locally convex spaces. It is our opinion that this technique can be exReceived by the editors September 23, 1974. AMS (MOS) subject classifications (1970). Primary 46A05, 46E10.

We present a result on the existence of Cournot-Nash equilibria in games with an atomless measure space of players each with nonordered preferences and with strategy sets in a separable Banach space. Our result dispenses with any convexity assumption on the preference correspondence. 1. Introduction. Recent work of Kim, Prikry and Yannelis (10, 11), Yan- nelis (15), and the authors (8) has shown the existence of Cournot-Nash equilibria in games with a measure space of players, each of whose strategy sets lives in a separable Banach space and each of whose pay-offs is generated by a nonordered relation. This work solved a problem left open in (7) and relied on (previously unknown) selection theorems of the Caratheodory type. For motivation to the problem, the reader can see (6 and 8). In this paper we present a result that dispenses with any kind of convexity as- sumption on the preference correspondence in games with an atomless measure space of players. Our result can be seen as an atomless analogue of the purely atomic result of Shafer and Sonnenschein (14), with the atomless hypothesis en- abling us to dispense with their convexity-type assumption in spite of the infinite- dimensional setting. Indeed, the theorem we present yields a new result even when it is specialized to a finite-dimensional Euclidean space (see Remark 2 below). As such, it generalizes the finite-dimensional results in Khan and Vohra (9). More- over, unlike (8, 10, and 15) there is no need for elaborate selection theorems of the Caratheodory type. Our new method of proof is suggested by (4) in which Grodal asked whether the assumptions of Theorem I in Khan-Vohra (9) were consistent. 2. Preliminary definitions and the result. Let (T, E,p) be a complete, finite measure space. Let E denote a separable Banach space over the real numbers R and E* its topological dual. The topology a(E,E*) will be referred to as the weak topology on E. The norm in E will be donoted by || • ||. P(E) will denote the hyperspace of all subsets of E including the empty set {0}. Pf(E) will denote the set of nonempty, closed subsets of E and Pk(E) the set of nonempty, compact subsets of E. Aw in front of f or k will mean that the

It is known that a lot of topological properties devolve from a basic space X X to the family M τ ( X ) {M_\tau }(X) of all τ \tau -smooth Borel measures endowed with the weak topology (or to certain subspaces of M τ ( X ) {M_\tau }(X) ). The aim of this paper is to show that among these topological properties there cannot be properties which are hereditary on closed subsets but not on countable products of X X , e.g. normality, paracompactness, the Lindelöf property, local compactness and σ \sigma -compactness. For this purpose it is proved that the countable product space X N {X^N} is homeomorphic to a closed subset of M τ ( X ) {M_\tau }(X) . A further consequence of this result is for example that, for the family M τ 1 ( X ) M_\tau ^1(X) of probability measures in M τ ( X ) {M_\tau }(X) , compactness, local compactness and σ \sigma -compactness are equivalent properties.

For families of τ \tau -smooth probability measures we give nonstandard characterizations of uniform τ \tau -smoothness, uniform tightness, uniform pretightness, and relative compactness in the weak topology. We apply these characterizations to obtain two important theorems of probability theory: A theorem of Topsøe and a theorem of Prohorov.

Let X X be a separable metrizable space. If K K is a compact space whose function space C ( K ) C(K) is weakly K \mathcal {K} -analytic, then the space C p ( X , K ) {C_p}(X,K) of continuous maps from X X to K K with the pointwise topology has the Lindelöf property. If E E is a Banach space whose weak topology is K \mathcal {K} -analytic, then each lower semicontinuous map from X X to the family of nonempty closed convex subsets of the unit ball of the dual E E with the weak*-topology admits a continuous selection. This extends some results of Corson and Lindenstrauss.

It is well-known that if T is an everywhere defined bounded operator on a Banach space X to a Banach space Y and T* is its adjoint, then the range R(T*) of T* is closed in X* if and only if the range R(T) of T is closed in Y (cf. [2, pp. 487-489]). The object of this note is to establish this result for closed but possibly unbounded operators. This result, for the unbounded case, is of great utility in the study of differential operators and has been considered by F. E. Browder,2 I. C. Gohberg and M. G. Kreln, and by G. C. Rota. In the sequel, we shall have occasion to consider a set under several different topologies. We shall use the following convention: if A is a linear set and B is a set of linear functionals on A, then the set A with the weak topology induced by the elements of B will be denoted by (A, B). Thus, the assertion that a set C is closed (dense, etc.) in (A, B) shall mean that C is a subset of A which is closed (dense, etc.) in this weak topology. If A is a Banach space, then the assertion that a set C is closed (dense, etc.) in A shall refer to the norm topology of A. Henceforth, X and Y will be Banach spaces. If T is a closed operator with domain D(T) in X and range R(T) in Y, then it was noted by Sz.-Nagy [5], that D(T) becomes a Banach space under the norm I XT = I X I + ITxI (which we shall call the T-norm) and T is a bounded operator on this space. If in addition, D(T) is dense in X, it is well known that T has a uniquely defined adjoint T* with domain D(T*) dense in (Y*, Y), and that T* is a closed operator.

Let X denote a locally compact Hausdorff space, C b ( X ) {C_b}(X) the space of all bounded continuous complex valued functions on X, and β \beta the strict topology for C b ( X ) {C_b}(X) . The separability of ( C b ( X ) , β ) ({C_b}(X),\beta ) is characterized in terms of X, albeit in a more general context. This result provides a negative answer to a conjecture made by Todd, contains the classical separability theorems for continuous function spaces, and relates to the concepts of τ \tau -smooth and tight functionals.

Let ⟨ E , E ′ ⟩ \langle E,E’\rangle be a dual pair of vector spaces. It is shown that whenever the weak and Mackey topologies on E E are different there is a nonconvex linear topology between them. In particular this provides a large class of nonconvex linear topologies having the Hahn Banach Extension Property.