Vaja Tarieladze

This monograph is devoted to some problems of the theory of probabilistic distributions in topological Abelian groups with sufficiently abundant dual groups.

A summable sequence (an) in a Banach space X is called lp-canonical, 1 < p < oo, if an = anven, n = 1,2 where ( a n ) € lp, v : lp —• X is a continuous linear operator and (en) is the natural basis of lp. We are showing that a summable sequence (an) in X is lp-canonical iff the operator u : c_0 —• X, with u_n = a_n, n = 1 , 2 , . . . is psumming. It follows that in a given Banach space X any summable sequence is lp-canonical iff any continuous linear operator from co to X is p-summing. The last assertion implies the following statement obtained previously in [Kva]: in a given Banach space X any summable sequence is lp-canonical for certain p, 2 < p < oo iff X does not contain \l_n^{infty} 's uniformly. For the spaces with a given cotype p we are obtaining the more precise results

showing, in particular, that in cotype 2 spaces any summable sequence is \l_2 - c a n o n i c a l , while in lp, with 2 < p < \infty, not any summable sequence is lp-canonical.

For a fixed sequence f. = (fn) of independent identically distributed symmetric random variables with , we introduce the notion of K^f.-convex Banach space and the notions of (fn)-bounding and (fn)-converging operators acting between Banach spaces. It is shown that the dual of the space of (fn)-converging operators between a Hilbert space and a K^f.-convex Banach space admits a precise description in terms of trace duality. The obtained results recover similar formulations for almost summing and γ-Radonifying operators.

We define GP-nuclear groups as topological Abelian groups for which the groups of summable and absolutely summable sequences are the same algebraically and topologically. It is shown that in the metrizable case only the algebraic coincidence of the mentioned groups is needed for GP-nuclearity. Some permanence properties of the class of GP-nuclear groups are obtained. Our final result asserts that nuclear groups in the sense of Banaszczyk are GP-nuclear. The validity of the converse assertion remains open.

In a previous paper ["Nuclear and GP-nuclear groups", Acta Math. Hungar. 88 (2000) 301--322] we introduced GP-nuclear groups as those topological Abelian groups for which the groups of summable and absolutely summable sequences are the same algebraically and topologically. It is proved in the present paper that, any countable direct sum of GP-nuclear groups is GP-nuclear and this may not hold in the uncountable case. At the end we show that a locally convex vector group is a GP-nuclear group if and only if it is nuclear. This recovers Grothendieck-Pietsch theorem for locally convex spaces. All necessary definitions and auxiliary assertions are presented in the text; in particular, we give a survey of some known results about GP-nuclear groups (including the Riemann-Dirichlet theorem for real numbers, the Dvoretzky-Rogers theorem for normed spaces, the results of W. Banaszczyk in "Additive subgroups of topological vector spaces [Lecture Notes Math. Springer 1466 (1991)] for nuclear groups and the related results in the above-mentioned paper of the authors). Thus, the paper is self-contained.

We study the topological semigroups that admit the adjunction of a non-isolated absorbing element and the structure and permanence properties of the class AA of topological semigroups admitting this type of adjunctions. No precompact topological group can belong to the class AA. More generally, a subsemigroup X of a compact Hausdorff semigroup K belongs to AA iff X misses the Suskevic kernel of the closure cl(X) in K. Every non-torsion abelian group belongs to AA when equipped with the discrete topology. Our interest in the class AA stems from the question of quasi-uniformizability of semigroups (every topological AA-group, after the adjunction of a non-isolated absorbing element, gives rise to a non-quasi-uniformizable semigroup).

It is natural to extend the Grothendieck Theorem on completeness, valid for locally convex topological vector spaces, to abelian topological groups. The adequate framework to do it seems to be the class of locally quasi-convex groups. However, in this paper we present examples of metrizable locally quasi-convex groups for which the analogue to Grothendieck Theorem does not hold. By means of the continuous convergence structure on the dual of a topological group, we also state some weaker forms of Grothendieck Theorem valid for the class of locally quasi-convex groups. Finally, we prove that for the smaller class of nuclear groups, BB-reflexivity is equivalent to completeness. 

It is shown that in an infinite-dimensional dually separated second category topological vector space X there does not exist a probability measure μ for which the kernel coincides with X. Moreover, we show that in “good” cases the kernel has the full measure if and only if it is finitedimensional. Also, the problem posed by S. Chevet [Kernel associated with a cylindrical measure, Springer-Verlag, 1981, p. 69] is solved by proving that the annihilator of the kernel of a measure μ coincides with the annihilator of μ if and only if the topology of μ-convergence in the dual space is essentially dually separated.

First, the basic properties of mean dilatation (MD-) numbers for linear operators acting from a finite-dimensional Hilbert space are investigated. Among other results, in terms of first and second order MD-numbers, a characterization of isometries is obtained and a dimension-free estimation of the p-th order MD-number by means of the first order MD-number is established. After that asymptotic MD-numbers for a continuous linear operator acting from an infinite-dimensional Hilbert space are introduced and it is shown that in the case of an infinite-dimensional domain the asymptotic p-th order MD-number, rather unexpectedly, is simply the p-th power of the asymptotic first order MD-number (Theorem 3.1).

Problems of the Mackey-continuity of characteristic functionals and the localization of linear kernels of Radon probability measures in locally convex spaces are investigated. First the class of spaces is described, for which the continuity takes place. Then it is shown that in a non-complete sigmacompact inner product space, as well as in a non-complete sigma-compact metizable nuclear space, there may exist a Radon probability measure having a non-continuous characteristic functional in the Mackey topology and a linear kernel not contained in the initial space. Similar problems for moment forms and higher order kernels are also touched upon. Finally, a new proof of the result due to Chr. Borell is given, which asserts that any Gaussian Radon measure on an arbitrary Hausdorff locally convex space has the Mackey-continuous characteristic functional

By means of a sequence ϕ· := (ϕn)n∈N of square-integrable functions a notion of a ϕ·-summing operator is defined. It is shown that if infn kϕnk2 > 0, then any ϕ·-summing operator is Gaussian-summing. This recovers a previously known result, which asserts the same in case when ϕ· := (ϕn)n∈N is an orthonormal sequence.

It is shown that any continuous from below exhaustive set function given on a σ-ring and with values in a regular topological space is continuous at ∅

We prove that a random series ∑ x i γ i , where (x i ) is a sequence of vectors in a Banach space and (γ i ) is a sequence of i.i.d Gaussian random variables is a.s. unconditionally convergent if and only if it is a.s. convergent and the series ∑ x i is unconditionally convergent. A similar statement in which (γ i ) is replaced by a sequence of independent random variables is proved also. We give some applications to Karhunen—Loève representations of Gaussian processes.

The property of Dunford-Pettis for a locally convex space was introduced by Grothendieck in 1953. Since then it has been intensively studied, with especial emphasis in the framework of Banach space theory. In this paper we define the Bohr sequential continuity property (BSCP) for a topological Abelian group. This notion could be the analogue to the Dunford-Pettis property in the context of groups. We have picked this name because the Bohr topology of the group and of the dual group plays an important role in the definition. We relate the BSCP with the Schur property, which also admits a natural formulation for Abelian topological groups, and we prove that they are equivalent within the class of separable metrizable locally quasi-convex groups. For Banach spaces (or for metrizable locally convex spaces), considered in their additive structure, we show that the BSCP lies between the Schur and the Dunford-Pettis properties.

Leaning on a remarkable paper of Pryce, the paper studies two independent classes of topological Abelian groups which are strictly angelic when endowed with their Bohr topology. Some extensions are given of the Eberlein–Šmulyan theorem for the class of topological Abelian groups, and finally, for a large subclass of the latter, Bohr angelicity is related to the Schur property.

Modifying appropriately the method of a forgotten work [1], we show that if a continuous mapping from a nonempty open subset U of a metrizable separable Baire topological vector space E to a locally convex space is directionally right differentiable on U along a comeager subset of E, then it is generically Gateaux differentiable on ˆ U. Our result implies that every metrizable separable Baire topological vector space is weak Asplund. It partially covers also the similar statements of [8, 16, 23].

The following results are announced: If a sequence of random elements tea normed space converges to zero in probability, then there exists a permutation of the given sequence, which is Cesáro convergent almost surely. Every bounded in L, sequence of random variables allows a permutation under which it satisfies the strong law of large numbers. A statement about Banach spaces with the Banach-Saks property is also given.

We give a survey of properties of weakly sub-Gaussian random elements in infinite-dimensional spaces. Some new results and examples are also given

The permutational and the weak permutational Banach-Saks properties are introduced and their relationship with the usual Banach-Saks and Weak Banach-Saks properties is clarified

Fréchet–Urysohn (briefly F-U) property for topological spaces is known to be highly non-multiplicative; for instance, the square of a compact F-U space is not in general Fréchet–Urysohn [P. Simon, A compact Fréchet space whose square is not Fréchet, Comment. Math. Univ. Carolin. 21 (1980) 749–753. [27]]. Van Douwen proved that the product of a metrizable space by a Fréchet–Urysohn space may not be (even) sequential. If the second factor is a topological group this behaviour improves significantly: we have obtained (Theorem 1.6(c)) that the product of a first countable space by a F-U topological group is a F-U space. We draw some important consequences by interacting this fact with Pontryagin duality theory. The main results are the following:

(1) If the dual group of a metrizable Abelian group is F-U, then it must be metrizable and locally compact.

(2) Leaning on (1) we point out a big class of hemicompact sequential non-Fréchet–Urysohn groups, namely: the dual groups of metrizable separable locally quasi-convex non-locally precompact groups. The members of this class are furthermore complete, strictly angelic and locally quasi-convex.

(3) Similar results are also obtained in the framework of locally convex spaces.

Another class of sequential non-Fréchet–Urysohn complete topological Abelian groups very different from ours is given in [E.G. Zelenyuk, I.V. Protasov, Topologies of Abelian groups, Math. USSR Izv. 37 (2) (1991) 445–460.

We introduce a notion of a Schwartz group, which turns out to be coherent with the well known concept of a Schwartz topological vector space. We establish several basic properties of Schwartz groups and show that free topological Abelian groups, as well as free locally convex spaces, over hemicompact k-spaces are Schwartz groups. We also prove that every hemicompact k-space topological group, in particular the Pontryagin dual of a metrizable topological group, is a Schwartz group.

We clarify the connection between: (1) regular conditional probabilities relative to a sigma-algebra, (2) regular conditional probabilities relative to a mapping and (3) disintegrations with respect to a mapping

It is shown that a Gaussian measure in a given infinite-dimensional Banach space always admits an essentially unique Gaussian disintegration with respect to a given continuous linear operator. This covers a similar statement made earlier in [Lee and Wasilkowski, Approximation of linear functionals on a Banach space with a Gaussian measure, J. Complexity 2(1) (1986) 12–43.] for the case of finite-rank operators.

We study various degrees of completeness for a Tychonoff space X. One of them plays a central role, namely X is called a Conway space if X is sequentially closed in its Stone–Chech compactification βX (a prominent example of Conway spaces is provided by Dieudonné complete spaces). The Conway spaces constitute a bireflective subcategory Conw of the category Tych of Tychonoff spaces. Replacing sequential closure by the general notion of a closure operator C, we introduce analogously the subcategory ConwC of C-Conway spaces, that turns out to be again a bireflective subcategory of Tych.We show that every bireflective subcategory of Tych can be presented in this way by building a Galois connection between bireflective subcategories of Tych and closure operators of Top finer than the Kuratowski closure. Other levels of completeness are considered for the (underlying topological spaces of) topological groups. A topological group G is sequentially complete if it is sequentially closed in its Ra˘ıkov completion ˜G. The sequential completeness for topological groups is stronger than Conway’s property, although they coincide in some classes of topological groups, for example: free (Abelian) topological groups, pseudocompact groups, etc.

Theorem 1 proves (among the others) that for a locally compact topological group X the following assertions are equivalent: (i) X is metrizable and σ-compact. (ii) Cp(X) is analytic. (iii) Cp(X) is K-analytic. (iv) Cp(X) is Lindelöf. (v) Cc(X) is a separable metrizable and complete locally convex space. (vi) Cc(X) is compactly dominated by irrationals. This result supplements earlier results of Corson, Christensen and Calbrix and provides several applications, for example, it easily applies to show that: (1) For a compact topological group X the Eberlein, Talagrand, Gul'ko and Corson compactness are equivalent and any compact group of this type is metrizable. (2) For a locally compact topological group X the space Cp(X) is Lindelöf iff Cc(X) is weakly Lindelöf. The proofs heavily depend on the following result of independent interest: A locally compact topological group X is metrizable iff every compact subgroup of X has countable tightness (Theorem 2). More applications of Theorem 1 and Theorem 2 are provided.

It is known that every continuous character on a topological vector

space can be lifted to a continuous linear functional and, moreover, these liftings give rise to a topological isomorphism between the dual group and the dual space, when both are endowed with the compact-open topology. We investigate the presence of these properties in more general topologized real vector spaces.

If (X,Q) is a quasi-uniform space, then in the hyperspace P0(X) of

all non-empty subsets of X we investigate the several quasi-uniformities related with the Bourbaki–Hausdorff quasi-uniformity ([5], [10], [11], [12]). We show that if (X,Q) is a quasi-uniform monoid (conoid), then P0(X) with respect to the corresponding algebraic operations and quasi-uniformities is again a quasiuniform

monoid (conoid). Moreover, it is demonstrated that if (X,Q) is a

quasi-uniform conoid, then in case of the hyperspace Pc(X) of all non-empty convex subsets of X the scalar multiplication on positive real numbers has some nice continuity properties.

In this paper first we present elements of a theory of integration of a quasi-uniform conoid valued functions with respect to a positive measure following the doctoral thesis of the first author "Integration on Uniform Type Conoids". Then we show that it is possible to apply this theory for integration of functions with values in the hyperspace of non-empty convex subsets of a given

quasi-uniform conoid. To realize such a possibility we treat the considered hyperspace as a quasi-uniform conoid in a rather natural way (as this is done in our joint work "Uniform type Hyperspaces").

Let (\xi_k, k\geq1) be a sequence of independent, identically distributed second order mean zero random elements in a separable Hilbert space H and A be an element of a certain class of linear continuous operators H\rightarrowH such that ||A||<1. We prove that if  ||I-A|| tends to zero, where I is the identity operator, then the normalized sum (I – A^2)^{1/2}\eta_A converges in distribution to a Gaussian random element

A topologized semigroup X having an evenly continuous resp., topologically equicontinuous, family RX of right translations is investigated. It is shown that: (1) every left semitopological semigroup X with an evenly continuous family R_X is a topological semigroup, (2) a semitopological group X is a paratopological group if and only if the family R_X is evenly continuous and (3) a semitopological group X is a topological group if and only if the family R_X is topologically equicontinuous. In particular, we get that for any paratopological group X which is not a topological group, the family R_X provides an example of a transitive group of homeomorphisms of X that is evenly continuous and not topologically equicontinuous. The last conclusion answers negatively a question posed by H.L. Royden.

The concepts of equicontinuity, even continuity, topological equicontinuity and the newly defined concepts of compact equicontinuity and compact topological equicontinuity are compared. It is shown that for a set of group homomorphisms from a semitopological group to a topological group all these notions of equicontinuities coincide. It is shown also that an infinite-dimensional normed space endowed with its weak topology is an example of a space which does not satisfy the Ascoli theorem in Noble's sense.

By means of a sequence ϕ =( ϕ_n), n\in N of square-integrable functions, not necessarily orthonormal, on a positive measure space (\Omega,\new) a notion of a ϕ⋅-summing operator can be defined. We show that the class \Pi_ϕ (X,Y) of all ϕ⋅-summing operators acting between Banach spaces X, Y is non-trivial iff ϕ⋅ is a 2-

dominated sequence. It is shown also that if ϕ⋅ is a 2-dominated sequence, then every 2-summing operator is ϕ⋅ *-summing

It is proved that a locally quasi-convex group is a Schwartz group if and only if every continuously convergent filter on its dual group converges locally uniformly. We also show that for metrizable separable groups a similar result remains true when filters are replaced by sequences. As an ingredient in the proofs of these results, we obtain a Schauder-type theorem on compact homomorphisms acting between the natural group analogues of normed spaces.

In a recent paper by D. Shakhmatov and J. Spěvák [D. Shakhmatov, J. Spěvák, Group-valued continuous functions with the topology of pointwise convergence, Topology Appl. 157 (2010) 1518–1540] the concept of a TAP group is introduced and it is shown in particular that NSS groups are TAP. We define the classes of STAP and HTAP groups and show that in general one has the inclusions NSS ⊂ STAP ⊂ HTAP ⊂ TAP. We show that metrizable STAP groups are NSS and that Weil-complete metrizable TAP groups are NSS as well. We prove that an abelian TAP group is HTAP, while, as recently proved by D. Dikranjan and the above mentioned authors, there are nonabelian metrizable TAP groups which are not HTAP. A remarkable characterization of pseudocompact spaces obtained in the above mentioned paper asserts: a Tychonoff space X is pseudocompact if and only if C_p(X,R) has the TAP property. We show that for no infinite Tychonoff space X, the group C_p(X,R) has the STAP property. We also show that a metrizable locally balanced topological vector group is STAP iff it does not contain a subgroup topologically isomorphic to Z^(N).

One variant of the transference lemma is proved. By its application we get a refined version of the Garsia inequality for orthogonal systems. Moreover, it is shown that the fulfillment of the “sigma-theta” condition for the Fourier series of a continuous periodic Banach space valued function implies the uniform convergence of a rearranged series to the function

The necessary and sufficient conditions for the universality of the complex series are established 

We call a Gaussian random element $\eta$ in a Banach space $X$ with a Schauder basis ${\bf e}=(e_n)$ diagonally canonical (for short, $D$-canonical) with respect to ${\bf e}$ if the distribution of $\eta$ coincides with the distribution of a random element having the form $B\xi$, where $\xi$ is a Gaussian random element in $X$, whose ${\bf e}$-components are stochastically independent and $B: X\to X$ is a continuous linear mapping. In this paper we show that if $X=l_p$, $1\le p<\infty$ and $p\neq2,$ or $X=c_0,$ then there exists a Gaussian random element $\eta$ in $X,$ which is not $D$-canonical with respect to the natural basis of $X$. We derive this result in the case when $X=l_p$, $2< p<\infty$, or $X=c_0$ from the following statement, an analogue which was known earlier only for Banach spaces without an unconditional Schauder basis: if $X=l_p,$ $2<p<\infty,$ or $X=c_0,$ then there exists a Gaussian random element $\eta$ in $X$ such that the distribution of $\eta$ does not coincide with the distribution of the sum of almost surely convergent in $X$ series $\sum_{n=1}^\infty{}x_ng_n$, where $(x_n)$ is an unconditionally summable sequence of elements of $X$ and $(g_n)$ is a sequence of stochastically independent standard Gaussian random variables.

Motivated from [31], call a precompact group topology τ on an abelian group G ss-precompact (abbreviated from single sequence precompact) if there is a sequence in G such that τ is the finest precompact group topology on G making converge to zero. It is proved that a metrizable precompact abelian group is ss-precompact iff it is countable. For every metrizable precompact group topology τ on a countably infinite abelian group G there exists a group topology η such that η is strictly finer than τ and the groups and have the same Pontryagin dual groups (in other words, is not a Mackey group in the class of maximally almost periodic groups).

We give a complete description of all ss-precompact abelian groups modulo countable ss-precompact groups from which we derive:

(1)No infinite pseudocompact abelian group is ss-precompact.

(2)An ss-precompact group G is a k-space if and only if G is countable and sequential.

(3)An ss-precompact group is hereditarily disconnected.

(4)An ss-precompact group has countable tightness.

We provide also a description of the sequentially complete ss-precompact abelian groups.

Motivated from [31], call a precompact group topology τ on an abelian group G ss-precompact (abbreviated from single sequence precompact) if there is a sequence in G such that τ is the finest precompact group topology on G making converge to zero. It is proved that a metrizable precompact abelian group is ss-precompact iff it is countable. For every metrizable precompact group topology τ on a countably infinite abelian group G there exists a group topology η such that η is strictly finer than τ and the groups and have the same Pontryagin dual groups (in other words, is not a Mackey group in the class of maximally almost periodic groups).

We give a complete description of all ss-precompact abelian groups modulo countable ss-precompact groups from which we derive:

(1)No infinite pseudocompact abelian group is ss-precompact.

(2)An ss-precompact group G is a k-space if and only if G is countable and sequential.

(3)An ss-precompact group is hereditarily disconnected.

(4)An ss-precompact group has countable tightness.

We provide also a description of the sequentially complete ss-precompact abelian groups.

A class G of maximally almost periodic (MAP) topological Abelian groups is called a UMAP-class if it has the following property: if G is an Abelian group and τ1 and τ2 are distinct group topologies in G such that (G, τ1) and (G, τ2) are in G, then (G, τ1) = (G, τ2). Several examples of UMAP-classes are discussed. In particular, it is shown that the class PMAP of all Polish MAP-groups

is a UMAP-class.

For a topological abelian group X we topologize the group c0(X) of all X-valued null sequences in a way such that when X=ℝ the topology of c0(ℝ) coincides with the usual Banach space topology of the classical Banach space c0. If X is a non-trivial compact connected metrizable group, we prove that c0(X) is a non-compact Polish locally quasi-convex group with countable dual group c0(X)_^. Surprisingly, for a compact metrizable X, countability of c0(X)_^∧ leads to connectedness of X.

Our principal application of the above results is to the class of locally quasi-convex Mackey groups ( LQC -Mackey groups). A topological group (G,μ) from a class g of topological abelian groups will be called a Mackey group in g or a g-Mackey group if it has the following property: if ν is a group topology in G such that (G,ν)and (G,ν) has the same character group as (G,μ), then ν≤μ.

Based upon the results obtained for c0(X), we provide a large family of metrizable precompact (hence, locally quasi-convex) connected groups which are not LQC -Mackey. Namely, we show that for a connected compact metrizable group X≠{0}, the group c0(X), endowed with the topology induced from the product topology on Xℕ, is a metrizable precompact connected group which is not a Mackey group in LQC.

Since metrizable locally convex spaces always carry the Mackey topology – a well-known fact from Functional Analysis –, our results prove that a Mackey theory for abelian groups is not a simple traslation of items known to hold for locally convex spaces. This paper is a contribution to the Mackey theory for groups, where properties of a topological nature like compactness or connectedness have an important r

It is shown that every infinite-dimensional real Banach space  contains a sequence, were its subsequence satisfies some editional properties. This result implies, in particular, that the rearrangement theorem and the Dvoretzky–Hanani theorem fail drastically for infinite-dimensional Banach spaces.

For a topological Abelian group X, we consider in the group X ^ℕ the uniform topology and study some properties of the obtained topological group. In particular, we show, that if X=S is the circle group, then the group S^N endowed with the uniform topology has the dual group of cardinality 2^c.

A Hausdorff locally convex topological vector space

X is called a dually c-Mackey space if its dual space X*

 equipped with the topology of uniform convergence on compact subsets of X is a Mackey space. It is shown that a metrizable locally

convex space X is a dually c-Mackey space iff X has the Schur

property. It follows that a reflexive Banach space X is a dually

c-Mackey space iff X is finite-dimensional.

In this paper a sufficient and a necessary condition

for unconditional convergence of a series in a Banach space with

an unconditional basis are analyzed.

For a topological vector space (X, τ ), we consider the family LCT (X, τ ) of all locally convex topologies defined on X, which give rise to the same continuous linear functionals as the original

topology τ . We prove that for an infinite-dimensional reflexive Banach space (X, τ ), the cardinality of LCT (X, τ ) is at least c.

We give a characterization of weakly subgaussian random elements that are γ-subgaussian in infinite-dimensional Banach and Hilbert spaces.

In this paper we obtain a result which implies, in particular that the sum range of a special quaternion series is a closed proper subfield of the division ring of quaternions, isometrically isomorphic to the field of complex numbers

A counterpart of the Mackey–Arens Theorem for the class of locally quasi-convex topological Abelian groups (LQC-groups) was initiated in Chasco et al. (Stud Math 132(3):257–284, 1999). Several authors have been interested in the problems posed there and have done clarifying contributions, although the main question of that source remains open. Some differences between the Mackey Theory for locally convex spaces and for locally quasi-convex groups, stem from the following fact: The supremum of all compatible locally quasi-convex topologies for a topological abelian group G may not coincide with the topology of uniform convergence on the weak quasi-convex compact subsets of the dual group G∧. Thus, a substantial part of the classical Mackey–Arens Theorem cannot be generalized to LQC-groups. Furthermore, the mentioned fact gives rise to a grading in the property of “being a Mackey group”, as defined and thoroughly studied in Díaz Nieto and Martín-Peinador (Proceedings in Mathematics and Statistics 80:119–144, 2014). At present it is not known—and this is the main open question—if the supremum of all the compatible locally quasi-convex topologies on a topological group is in fact a compatible topology. In the present paper we do a sort of historical review on the Mackey Theory, and we compare it in the two settings of locally convex spaces and of locally quasi-convex groups. We point out some general questions which are still open, under the name of Problems.

For a normed infinite-dimensional space, we prove that the family of all locally convex topologies which are compatible with the original norm topology has cardinality greater or equal to c 

Let E and F be topological vector spaces and let G and Y be topological abelian groups. We say that E is sequentially barrelled with respect to F if every sequence (u_n)n\in N of continuous linear maps from E to F which converges pointwise to zero is equicontinuous. We say that G is barrelled with respect to F if every set H of continuous homomorphisms from G to F, for which the set H(x) is bounded in F for every x\in E, is equicontinuous. Finally, we say that G is g-barrelled with respect to Y if every H\subsetHom(G,Y) which is compact in the product topology of Y^G is equicontinuous. We prove that

• a barrelled normed space may not be sequentially barrelled with respect to a complete metrizable locally bounded topological vector space,
• a topological group which is a Baire space is barrelled with respect to any topological vector space,
• a topological group which is a Namioka space is g-barrelled with respect to any metrizable topological group,
• a protodiscrete topological abelian group which is a Baire space may not be g-barrelled (with respect to R/Z).

We also formulate some open questions.

We give a short survey concerning sub-Gaussian random elements in a Banach space and prove a statement about the induced operator of a bounded random element in a Hilbert space

Let X be a topological abelian group and H a subgroup of X. We find conditions under which local quasi-convexity of both H and X/H results in the same property for X. This is true for instance if H is precompact, or if X is metrizable and H is a dually embedded subgroup which is also either discrete or bounded torsion. We also give some general principles and point out some errors we have found in the existing literature on this problem.

We show that the following general version of the Riemann–Dirichlet theorem is true: if every rearrangement of a series with pairwise commuting terms in a Hausdorff topologized semigroup converges, then its sum range is a singleton

We discuss the validity of Menshov-Rademacher theorem for quasi-orthogonal series introduced by D.A. Menshov in a paper published in 1927 and for quasi-orthogonal sequence considered in a paper of 1948 by M. Kac, R. Salem and A. Zygmund.

Criteria for almost sure boundedness and convergence to zero almost surely of Banach space valued independent Gaussian random elements are found. The obtained statements can be viewed as vector-valued versions of the corresponding results due to N. Vakhania. Moreover, from the obtained statements a strong law of large numbers is derived in the form of Yu. V. Prokhorov.

The permutational and the weak permutational Banach-Saks properties are introduced and their relationship with the usual Banach-Saks and Weak Banach-Saks properties is clarified

It is shown that the rearrangement theorem is not true for infinite-dimensional Banach spaces.