On compacta which are $l$-equivalent to $I^{n}$

Minimal cusco maps have applications in functional analysis, in optimization, in the study of weak Asplund spaces, in the study of differentiability of functions, etc. It is important to know their topological properties. Let X be a Hausdorff topological space, MC(X) be the space of minimal cusco maps with values in R and ?UC be the topology of uniform convergence on compacta. We study completemetrizability and cardinal invariants of (MC(X), ?UC). We prove that for two nondiscrete locally compact second countable spaces X and Y, (MC(X), ?UC) and (MC(Y), ?UC) are homeomorphic and they are homeomorphic to the space C(Ic) of continuous real-valued functions on Ic with the topology of uniform convergence.

The Approximation Property for Certain Spaces of Holomorphic Mappings

We prove that a compact space embeds into a ‐product of compact metrizable spaces (‐product of intervals) if and only if is (strongly countable‐dimensional) hereditarily metalindelöf and every subspace of has a nonempty relative open second countable subset. This provides novel characterizations of ‐Corson and compact spaces. We give an example of a uniform Eberlein compact space that does not embed into a product of compact metric spaces in such a way that the ‐product is dense in the image. In particular, this answers a question of Kubiś and Leiderman. We also show that for a compact space , the property of being compact is determined by the topological structure of the space of continuous real‐valued functions of equipped with the pointwise convergence topology. This refines a recent result of Zakrzewski.

Let {mathbb {K}} be a non-trivially valued non-Archimedean complete field. Let ell _{infty }({mathbb {N}}, {mathbb {K}}) [ell _c({mathbb {N}}, {mathbb {K}});c_0({mathbb {N}}, {mathbb {K}})] be the space of all sequences in {mathbb {K}} that are bounded [relatively compact; convergent to 0] with the topology of pointwise convergence (i.e. with the topology induced from {mathbb {K}}^{{mathbb {N}}}). Let X be an infinite ultraregular space and let C_p(X,{mathbb {K}}) be the space of all continuous functions from X to {mathbb {K}} endowed with the topology of pointwise convergence. It is easy to see that C_p(X,{mathbb {K}}) is metrizable if and only if X is countable. We show that for any X [with an infinite compact subset] the space C_p(X,{mathbb {K}}) has an infinite-dimensional [closed] metrizable subspace isomorphic to c_0({mathbb {N}}, {mathbb {K}}). Next we prove that C_p(X,{mathbb {K}}) has a quotient isomorphic to c_0({mathbb {N}}, {mathbb {K}}) if and only if it has a complemented subspace isomorphic to c_0({mathbb {N}}, {mathbb {K}}). It follows that for any extremally disconnected compact space X the space C_p(X,{mathbb {K}}) has no quotient isomorphic to the space c_0({mathbb {N}}, {mathbb {K}}); in particular, for any infinite discrete space D the space C_p(beta D, {mathbb {K}}) has no quotient isomorphic c_0({mathbb {N}}, {mathbb {K}}). Finally we investigate the question for which X the space C_p(X,{mathbb {K}}) has an infinite-dimensional metrizable quotient. We show that for any infinite discrete space D the space C_p(beta D, {mathbb {K}}) has an infinite-dimensional metrizable quotient isomorphic to some subspace ell _c^0({mathbb {N}}, {mathbb {K}}) of {mathbb {K}}^{{mathbb {N}}}. If {mathbb {K}} is locally compact then ell _c^0({mathbb {N}}, {mathbb {K}})=ell _{infty }({mathbb {N}}, {mathbb {K}}). If |n1_{{mathbb {K}}}|ne 1 for some nin {mathbb {N}}, then ell _c^0({mathbb {N}}, {mathbb {K}})=ell _c ({mathbb {N}}, {mathbb {K}}). In particular, C_p(beta D, {mathbb {Q}}_q) has a quotient isomorphic to ell _{infty }({mathbb {N}}, {mathbb {Q}}_q) and C_p(beta D, {mathbb {C}}_q) has a quotient isomorphic to ell _c({mathbb {N}}, {mathbb {C}}_q) for any prime number q.

for t E T and s E S. This definition can be restated in equivalent ways in terms of particular flows. A flow (X, T) is a transformation group [8] where X is a topological space and Tis the additive group of the reals. We designate the action of t E T on x eX by x x,. A point xeX is said to be almost periodic if for each neighborhood U of x there is a relatively dense set S c T such that x. E U if s E S. We let B be the Banach space of uniformly continuous bounded real valued functions of the reals. The norm of an element of B shall be the supremum of its absolute values. We let (B, T) be the flow defined by x -) x, where x,(s) = x(t + s). Uniformly almost periodic functions may be defined in any one of the three following ways. (1) A function x in B is uniformly almost periodic if x is an almost periodic point of (B, T). (2) A function x in B is uniformly almost periodic if its orbit closure in (B,T) is compact. (3) A function x in B is uniformly almost periodic if it may be extended to a continuous function on the Bohr compactification of the reals. The fact that these three statements are equivalent may be found in [3; 12]. For many notions of functions being recursive under translation the compactopen topology is more natural than the topology of uniform convergence. With this in mind we let C be the set of continuous real valued functions of the reals with the compact open topology. This topology is equivalent to the topology induced by the metric

In [1], we proved a criterion for the disjointness of two induced representations UL and UM of a Lie group G, where L and M are finite-dimensional unitary representations of compact subgroups H and K, respectively, of G. The purpose of this paper is to improve this theorem by getting a stronger conclusion, while dropping the conditions that G be a Lie group arld H be compact, and that L and M be finite-dimensional. Moreover, the restriction on K is weakened to read: G has arbitrarily small neighborhoods of the identity invariant under the adjoint action of K on G. Finally, the proof given below is fairly elementary, while the proof in [I ] is quite involved. Notations and conventions: Let Al be a topological vector space. C(G, CU) will denote the space of continuous functions from G to cA, equipped with the topology of uniform convergence on compact subsets of G while Co(G, CLL) will denote the space of those fC-C(G, CLL) with compact support. If cAl is omitted it is understood that ca=C. If cUl is another topological vector space, ?(ca, cU4) will denote the space of continuous linear maps from cA into cal equipped with the topology of bounded convergence. All integrations are with respect to right Haar measure. For any locally compact group G, G will denote its modular function. If f, g Co (G), f o g will denote the convolution of f and g, and f* is defined by f *(x) = c G(x) -f(x-1)If L and M are representations of G, R(L, M) will denote the space of intertwining operators for L and M (see [3]), while I(L, M) will denote the dimension of R(L, M). For the definition of induced representation used below, see [1]. Finally, for any function f on the group G and any xGG, fx and fx are defined by fx(y) =f(x-ly) and fx(y) =f(yx).

The topologies induced by two families of seminorms on a vector space of functions g: R + x R + x El -* El are compared. It is found that the continuous dependence of solutions of the Volterra equation x(t) f(t) + 5 , g(t, s, x(s)) ds does not hold for the weaker topology. This result corrects an error in the book of Miller, Benjamin, Menlo Park, Calif., 1971. The purpose of this paper is to give a counterexample to a proposition of R. K. Miller, concerning continuous dependence of solutions of Volterra integral equations. Consider the nonlinear Volterra equation x(t) = f(t) + f g(t, s, x(s)) ds, (E) where f E C(R +; E) (that is, f is a continuous function from R + = [0, oo) to the n-dimensional Euclidean space) and g is a function defined on R + X R + x En with values in En. In his book [2] Miller defines a vector space g of functions g: R + x R + x En--> En which satisfy certain hypotheses. Then he defines a family of seminorms on g (see (2) below) and proves, roughly speaking, that the solutions x of (E) depend continuously on the functionsf and g, where the functionsf belong to C(R +; En) topologized with the topology of uniform convergence on compact subsets and the functions g belong to 9 topologized with the locally convex vector topology induced by the seminorms mentioned above (cf. Theorem II.4.2 and its corollaries [2, pp. 108-116]). Miller gives an alternative characterization of the seminorms (see (4) below). In this note I will prove that this characterization is erroneous and that the seminorms on the right side of (4) lead to a topology on g for which the continuity theorems in [2] do not hold. The exact definition of g is given in [2, pp. 106-108]. For the purpose of this note it suffices to know that 9 contains all functions of the type g(t, s, x) = a(t s)h(s, x), (1) where h is a continuous function R + x En---En, and a is a locally integrable real valued function which vanishes on (oo, 0). The seminorms on 9 are defined in the following way (cf. [2, p. 107]): For every g E 9, every positive real number K and every bounded set B in Et, p(g, K, B) is defined by p(g, K, B) = sup{f Ig(t, s, p(s))I ds|t E[0, K], q E C([0, K]; B)}, (2) Received by the editors May 9, 1980. AMS (MOS) subject classifications (1970). Primary 45D05.

We establish relationships between various topological selection games involving the space of minimal cusco maps into the real line and the underlying domain. These connections occur across different topologies, including the topology of pointwise convergence and the topology of uniform convergence on compacta. Full and limited-information strategies are investigated. The primary games we consider are Rothberger-like games, generalized point-open games, strong fan-tightness games, Tkachuk’s closed discrete selection game, and Gruenhage’s W-games. We also comment on the difficulty of generalizing the given results to other classes of functions.

We prove that the additive group (E*, τ k (E)) of an $${\mathcal {L}_\infty}$$ -Banach space E, with the topology τ k (E) of uniform convergence on compact subsets of E, is topologically isomorphic to a subgroup of the unitary group of some Hilbert space (is unitarily representable). This is the same as proving that the topological group (E*, τ k (E)) is uniformly homeomorphic to a subset of $${\ell_2^\kappa}$$ for some κ. As an immediate consequence, preduals of commutative von Neumann algebras or duals of commutative C*-algebras are unitarily representable in the topology of uniform convergence on compact subsets. The unitary representability of free locally convex spaces (and thus of free Abelian topological groups) on compact spaces, follows as well. The above facts cannot be extended to noncommutative von Neumann algebras or general Schwartz spaces.

Minimal usco/cusco maps are very important in functional analysis, optimization, selection theorems, the study of differentiability of Lipschitz functions, etc. We study topologies of uniform convergence on bornologies on the space of minimal usco and minimal cusco maps. We find sufficient conditions for metrizability and complete metrizability of minimal usco and minimal cusco maps equipped with the topology of uniform convergence on bornologies. We also investigate Fréchet locally convex subspaces of these spaces.

Let E be the set of equations in the hull of a fixed two-dimensional, almost-periodic, linear differential equation with interval spectrum which admit a bounded solution.Among other things, we prove that is of first category and of measure zero. Introduction. In [8],[9], Sacker and Sell defined the spectrum of a linear, nonautonomous differential system x = B(t)x.In [2], it was proved that a two-dimensional example of Millionscikov [4] with almost-periodic coefficients has interval spectrum.Our purpose here is to prove several propositions concerning two-dimensional linear systems with uniquely ergodic hull (2.1 and 2.2 below) which have interval spectrum.In particular, we prove a result which implies the following statement.Let denote the hull of the system, let fig be the unique ergodic measure on , and let the spectrum of the system be [ -, ] for some > 0 (the spectrum may be made symmetric about zero by a simple normalization).Let 0 = {w : the equation defined by w (see 2.1 and 2.3) admits a bounded solution).By the definition of spectrum and [8, Theorem 3], 0 is nonempty; we prove that Pq(10) = 0, and that 0 is of first category in .2. Preliminaries.We introduce notation and review some definitions and results.2.1.Definitions.Let C be the space of all continuous mappings from R to the set of 2 X 2 real matrices.Give C the topology of uniform convergence on compact sets.The map $:CxR-*C:(A, t) -> A,, where A,(s) = A(t + s), defines a real flow [1] on C. Suppose B G C is uniformly bounded and uniformly continuous.Then = cls{7?,:t G R} c C is compact metric, and $|o,xR defines a flow (, R).We can "extend B to " as follows: let b(w) = w(0) (u G ); then b(ut) = w,() = u(t) (w G , / G R).In particular, if w0 = B G , then b(u0 t) = B(t).We call the hull of B.2.2.Definitions, notation.Let (A", R) be a flow, where X is a compact metric space.We denote the "position" of x G X after "time" t GRby x t.Say (X, R) is minimal if the orbit {x t: t GR) is dense in X for all x G X. Say (X, R) is uniquely ergodic if there is exactly one measure p on X which is invariant with respect to (A', R).Here and below, "measure" means "Radon probability measure".See [1], [7].2.3.Notation.Let B, , b be as in 2.1 (thus B is a uniformly bounded and uniformly continuous, 2 X 2-matrix valued function on R).

It is proved that any completely regular topological space is determined up to homeomorphism by the topological lattice (with the topology of pointwise convergence) of all its continuous real-valued functions. The well-known result of Kaplansky (for compact spaces) is a corollary of this theorem.

The latter defines the usual equivalence relation in Moo. We denote by f the equivalence class of each feM ~ with respect to this equivalence relation. We say that a set F carries p ifF~ and p(E-F)=O. In the case when E is a compact space and/~ a positive Radon measure on E, we denote by Supp ~ the smallest (= intersection) of all closed sets F c E, carrying p. If f: E~R and AcE, we denote by fla the restriction of f to A; finally if HcMoo we use the notation Hla for the set of all hlA, with h~H. The author is indebted to Professors Dietrich K61zow and John C. Oxtoby for several helpful remarks and relevant comments concerning the contents of this paper. We begin with the following simple but very useful result. Proposition 1. Let H c M ~ be a set which is bounded and such that the relations h 1 ~H, h 2 ~H and h I 4: h 2 imply h 1 + h 2 . Assume that H is sequentially compact for the topology of pointwise convergence on E, that is for any sequence (h,) of elements of H, there is a subsequence (h,k) and an h~H such that h,k ~ h pointwise on E. Then H is compact and metrizable for the topology of pointwise convergence on E. Proof We consider on H the topology -[-1 of mean Ll-convergence 1, and the topology T of pointwise convergence on E. We divide the proof into two steps: I) We note first that (H, T1) is compact. In fact, for any sequence (h,) in H, there is a subsequence (h,~) and an h~H such that h,~ ~ h pointwise on E. By the Lebesgue dominated convergence theorem, we also have

A large class of function spaces, under the topology of pointwise convergence, are shown to be of first category.

Let T T be a topological space admitting a compatible proper metric, that is, a locally compact, separable and metrizable space. Let M T \mathcal {M}^T be the non-empty set of all proper metrics d d on T T compatible with its topology, and equip M T \mathcal {M}^T with the topology of uniform convergence, where the metrics are regarded as functions on T 2 T^2 . We prove that the set A T , 1 \mathcal {A}^{T,1} of metrics d ∈ M T d\in \mathcal {M}^T for which the Lipschitz-free space F ( T , d ) \mathcal {F}(T,d) has the metric approximation property is a dense set in M T \mathcal {M}^T , and is furthermore residual in M T \mathcal {M}^T when T T is zero-dimensional. We also prove that if T T is uncountable then the set A f T \mathcal {A}^T_f of metrics d ∈ M T d\in \mathcal {M}^T for which F ( T , d ) \mathcal {F}(T,d) fails the approximation property is dense in M T \mathcal {M}^T . Combining the last statement with a result of Dalet, we conclude that for any ‘properly metrizable’ space T T , A f T \mathcal {A}^T_f is either empty or dense in M T \mathcal {M}^T .