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It is an open question whether every derivation of a Fréchet GB$^{\ast }$-algebra $A[{\it\tau}]$ is continuous. We give an affirmative answer for the case where $A[{\it\tau}]$ is a smooth Fréchet nuclear GB$^{\ast }$-algebra. Motivated by this result, we give examples of smooth Fréchet nuclear GB$^{\ast }$-algebras which are not pro-C$^{\ast }$-algebras.

In this paper we define bornological and b-bornological convergence in locally convex cones. We give some characterizations for these convergences. In special case of locally convex topological vector spaces, both these concepts reduce to the known concept of Mackey-convergence. We investigate the relationship between bornological cones and bornological convergence. Finally, we investigate bornological completeness.

Using the concept of a topological affine space, it is proved that a partially ordered topological linear spaces associated with relatively uniform and order convergence can be represented by an inductive limit of its subspaces.

A Lebesgue-type integration theory in complete bornological locally convex topological vector spaces was introduced by the first author in [17]. In this paper we continue developing this integration technique and formulate and prove some theorems on integrable functions as well as some convergence theorems. An example of Dobrakov integral in non-metrizable complete bornological locally convex spaces is given.

We propose a definition for strict inductive limits in locally convex cones. By this definition, we prove that the strict inductive limit of a sequence of locally convex cones with the strict separation property has the same strict separation property. Also we establish that the strict inductive limit of a sequences of separated cones is separated too. Finally we verify barreledness for this strict inductive limit.

The bornological product measures via the generalized Dobrakov integral in complete bornological locally convex spaces are studied using the domination of considered vector measures. A Fubini-type theorem for such product measures is proven.

An inductive locally convex limit of reflexive topological spaces is reflexive iff it is almost regular.

Spaces of entire functions of several complex variables occupy an important position in view of their vast applications in various branches of mathematics, for instance, the classical analysis, theory of approximation, theory of topological bases etc. With an idea of correlating entire functions with certain aspects in the theory of basis in locally convex spaces, we have investigated in this paper the bornological aspects of the space $X$ of integral functions of several complex variables. By $Y$ we denote the space of all power series with positive radius of convergence at the origin. We introduce bornologies on $X$ and $Y$ and prove that $Y$ is a convex bornological vector space which is the completion of the convex bornological vector space $X$.

Any inductive limit of bornivorously webbed spaces is sequentially complete iff it is regular.

The criteria for non-containment of $l^1$ for the classes of Banach and Fréchet spaces are extended to the class of locally complete locally convex spaces the bounded sets of which are metrizable.

We show that the Gelfand-Mazur theorem holds for countable inductive limits of Frechet algebras (we do not assume that the homomorphisms which define the inductive limit are continuous, or one-to-one). This question is motivated by the fact that the spectrum of some elements of such an algebra may be empty. We also discuss in detail a countable inductive limit of Frechet algebras of holomorphic functions, which provides an elementary, but seminal, counterexample to the biinvariant subspace problem for complete, reflexive, locally convex spaces.

A generalization of I. Dobrakov's integral to complete bornological locally convex spaces is given.

A family $\{A_{\alpha }:\alpha \in \mathbb{N}^{\mathbb{N}}\}$ of sets covering a set $E$ is called a resolution for $E$ if $A_{\alpha }\subseteq A_{\beta }$ whenever $\alpha \leq \beta $. A locally convex space (lcs) $E$ is said to belong to class $\mathfrak{G}$ if there is a resolution $\{A_{\alpha }:\alpha \in \mathbb{N}^{\mathbb{N}}\}$ for $(E^{\prime},\sigma (E^{\prime },E))$ such that each sequence in any $A_{\alpha }$ is equicontinuous. The class $\mathfrak{G}$ contains `almost all' useful locally convex spaces (including $(LF)$-spaces and $(DF)$-spaces). We show that$\;\left( i\right) $ every semi-reflexive lcs $E$ in class $\mathfrak{G}$ is a Lindelof $\Sigma $-space in the weak topology (this extends a corresponding result of Preiss-Talagrand for WCG Banach spaces) and the weak* dual of $E$ is both $K$-analytic and has countable tightness, $\left( ii\right) $ a barrelled space $E$ has a weakly compact resolution if and only if $E$ is weakly $K$-analytic, and $\left( iii\right) $ if $E$ is barrelled or bornological then $E^{\prime }$ has a weak* compact resolution if and only if it is weak* $K$-analytic. As an additional consequence we provide another approach to show that the weak* dual of a quasi-barrelled space in class $\mathfrak{G}$ is $K$-analytic. These results supplement earlier work of Talagrand, Preiss, Cascales, Ferrando, Kąkol, Lopez Pellicer and Saxon.

Polynomially bornological, polynomially barrelled and polynomially infrabarrelled locally convex spaces over a non-Archimedean non-trivially valued complete field of characteristic zero are introduced and studied.

A notion of an almost regular inductive limits is introduced. Every sequentially complete inductive limit of arbitrary locally convex spaces is almost regular.

We prove the existence of an order-preserving function on a class of preordered topological spaces that are inductive limits of preordered spaces. Some applications to economics are given.

Lebesgue’s differentiation theorem (LDT) states that every monotonic real function is differentiable a.e. We investigate the validity of this theorem for functions with values in topological vector lattices. It is shown that a Fréchet lattice satisfies (LDT) iff it is isomorphic to a generalized echelon space, a Banach lattice satisfies (LDT) iff it is isomorphic to some l 1 ( Γ ) {l^1}\left ( \Gamma \right ) .

The object of this paper is to study the stability of the quasi-completeness of theLα spaces for locally convex vector valued functions, by the inductive limits and the countable products.

Let $\mathbb K$ be a complete non-Archimedean valued field and let $C(X,E)$ be the space of all continuous functions from a zero-dimensional Hausdorff topological space $X$ to a non-Archimedean Hausdorff locally convex space $E$. We will denote by $C_{b}(X,E)$ (resp. by $C_{rc}(X,E)$) the space of all $f\in C(X,E)$ for which $f(X)$ is a bounded (resp. relatively compact) subset of $E$. The dual space of $C_{rc}(X,E)$, under the topology $t_{u}$ of uniform convergence, is a space $M(X,E')$ of finitely-additive $E'$-valued measures on the algebra $K(X)$ of all clopen , i.e. both closed and open, subsets of $X$. Some subspaces of $M(X,E')$ turn out to be the duals of $C(X,E)$ or of $C_{b}(X,E)$ under certain locally convex topologies.\\ In this paper we continue with the investigation of certain subspaces of $M(X,E')$. Among other results we show that, if $E$ is a polar Fréchet space, then :\\ 1. The space $\mathcal{M}_{\theta_{o}}(X,E')$, of all $m\in M(X,E')$ for which the support of the corresponding measure $m^{\beta_{o}}$, on the Banaschewski compactification of $X$, is contained in the $\theta_{o}$-repletion of $X$, is complete under the topology of uniform convergence on the family $\mathcal{E}$ of all equicontinuous subsets $B$ of $C(X,E)$ for which $B(x)$ is a compactoid subset of $E$ for all $x\in X$.\\ 2. The space $M_{bs}(X,E')$, of all the so called strongly-separable members of $M(X,E')$ is complete under the topology of uniform convergence on the family of all uniformly bounded members of $\mathcal{E}$.\\ 3. The space $M_{s}(X,E')$ of all $m\in M(X,E')$, for which $ms$ is separable for all $s\in E$, is complete under the topology of uniform convergence on the family of all $B \in \mathcal{E}$ for which the set $B(X)$ is compactoid.

A Fréchet space is called twisted if it is not isomorphic to a countable product of Fréchet spaces with continuous norms. It is easy to show that no non-archimedean Fréchet space with a Schauder basis is twisted. We construct examples of non-archimedean twisted nuclear Fréchet spaces.

In the introductory Section 0. some of the most important points in the professional curriculum vitae of Jean Schmets are given. Then Section 1 is devoted to present (part of) the work for which Schmets was very well known until 1990: He has been the leading specialist in the world for locally convex spaces $C(X)$ of continuous functions with various topologies and for the corresponding spaces $C(X,E)$ of vector valued continuous functions. Finally, in Section 2 some results which Schmets obtained in cooperation with Manuel Valdivia since 1990 are reviewed: work on domains of real analytic existence, continuous linear right inverses for $C^\infty$- functions and Whitney extensions for non quasianalytic functions.

In this short note we prove that the locally convex inductive limit of commutative locally m-convex algebras is already locally m-convex whenever multiplication is continuous. A topological algebra is an (associative) algebra endowed with a vector space topology such that multiplication is jointly continuous. If one considers inductive limits (in the category of topological vector spaces) of topological algebras multiplication may fail to be continuous since it is a bilinear map on the product which is certainly not linear, and forming inductive limits only respects the linear structure (although this is quite clear, the dierence between linearity and bilinearity in this context has been overlooked several times in the literature). The most important category of topological algebras is that of locally m-convex (l.m.c.) algebras having a 0-neighbourhood basis consisting of m-convex sets (i.e. absolutely convex sets B which are multiplicative in the sense that B 2 B). This class was introduced by Michael [9] and Arens [2] since it allows generalizations and applications of the theory of Banach algebras. Since their work a number of authors studied the question whether the inductive limit (in the category of locally convex spaces) of l.m.c. algebras is again l.m.c., see e.g. [1, 3, 4, 5, 7, 8, 10]. This is indeed true for countable inductive limits of normed algebras [1, 4]. On the other hand, Warner [10] gave an example of an inductive limit of metrizable l.m.c. algebras which is not l.m.c., and in [4] it is shown that [

Let $Q$ be a bounded, convex and locally closed subset of \ $\C^N$, let $H(Q)$ be the space of all functions which are holomorphic on an open neighborhood of $Q$. We endow $H(Q)$ with its projective topology. We show that the topology of the weighted inductive limit of Fr\'echet spaces of entire functions which is obtained as the Laplace transform of the strong dual to $H(Q)$ can be described be means of canonical weighted seminorms if and only if the intersection of $Q$ with each supporting hyperplane to the closure of $Q$ is compact. We also find conditions under which this (LF)-space of entire functions coincides algebraically with its projective hull.

We characterize the Frechet-Montel (respectively, Fr6chet-Schwartz) spaces by sequences in their dual spaces. It is well known that a Frechet space E is Montel (respectively, Schwartz) if and only if it is separable and every weak*-null sequence in E' also converges strongly to zero (respectively, converges uniformly to zero on some zeroneighbourhood in E). Because of the Josefson-Nissenzweig theorem [3], Jarchow asked in [5] if these two results are true without the assumption of the separability condition on the Frechet space E. In this short note we give a positive answer to both of these questions. These results were discovered during the Conference of Function Theory on InfiniteDimensional Spaces held in the Universidad Complutense de Madrid from December 11-16, 1989. The Schwartz case had already been discovered independently by Lindstr6m and Schlumprecht in [7] and by Bonet in [1]. We start by recalling the following result from [7], which is based on a very deep result due to Schlumprecht [8]. Theorem 1. Let E be a Frechet space such that every weak*-null sequence in E' is also strongly null-convergent. Then E is reflexive. Now we are ready to prove the announced results. Theorem 2. A Frechet space E is Montel (respectively, Schwartz) if and only if weak*-null sequence in E' is also strongly null-convergent (respectively, converges uniformly to zero on some zero-neighbourhood in E). Proof. Only the Montel case needs a proof because every Frechet-Montel space is separable. If E is Montel then the condition is clearly necessary. Let us now assume that every weak*-null sequence in E' is strongly nullconvergent. By the theorem above [7] E must then be reflexive. Now we present two different proofs to show that E is Montel. First, by [6, Remark I] it is enough to show that UO is a(E', E)-sequentially compact for every zero-neighbourhood U in E. To do this observe that Received by the editors January 25, 1991. 1991 Mathematics Subject Classification. Primary 46A04, 46A1 1. ( 1993 American Mathematical Society 0002-9939/93 $1.00 + $.25 per page

Definitions and notations. Let T be a Tihonov space, C(T) = { f l f : T~R a n d f i s a continuous function}, C*(T)= {flfEC(T) and f i s a bounded function}, Cc (T) = {fI fEC(T) and f has a compact support}. If fEC(T) then Z(f)--{tETlf(t)=O}, Zc(f)=TN.Z(f) . X(T) is a vector lattice iff X(T)cC(T), f, gEX(T) and LER imply fVgEX(T), fAgEX(T), f +gEX(T), ~fEX(T). Let 1"1, T~ be Tihonov spaces, X(T1), Y(T2) two vector lattices, cp: X(TO~ -~ Y(T.,) is said to be a linear lattice homomorphism iff ~o is a lattice homomorphism and xl,x~EX(T) implies ~o(x~-x~)=cp(xO-cp(x~). X(T) is said to be a completely regular vector lattice over Tiff, whenever F is a closed set and t is a point in its complement, there exists a function fEX(T) such that 0~f_

In topological linear spaces convex hulls of bounded sets are, in general, not bounded. The question arises whether there is at least for every bounded set B a sequence {λν|νeℕ} of strictly positive numbers such that the set ∪{Σ l n λ v B|neℕ} is bounded. When this obtains, bounded sets share several of the properties known in locally convex spaces. The main result of this note is an example of a countable inductive limit of complete metrizable topological linear spaces which is neither regular nor sequentially complete and also fails to have the above “bounded summability property”.

A subspace G of a locally convex space E has property (b) if for every bounded set B of E the codimension of G in the linear hull of G ∪ B is finite, (5). Extending the results of (5) and (14), we prove that, if the strong dual of E is complete, then subspaces with property (b) inherit the following properties of E: σ-evaluability, evaluability, the property of being Mazur, semibornological and bornological. We also prove that a dense subspace with property (b) of a Mazur space is sequentially dense, and of a semibornological space – dense in the sense of Mackey (locally dense, following M. Valdivia).

In [3], H. Lang answers (a) affirmatively in case X is a Fr6chet space and all products .P~Pj, i#j, are compact. This generalizes a similar result by L. Svensson [4] for reflexive Banach spaces. The aim of this paper is to answer question (b). In fact we will prove that if Xis a Hausdorff locally convex topological vector space and PiPj is compact for i#j, then 3/1 +... +Nm is complemented. Moreover a continuous linear projection onto this sum is given by P1 +... +Pm, modulo compact operators. I f X is a Hilbert space, we will prove that NI +... +N,, is closed ff Nt +N~ is closed for all i, j and every product "PiPjPk is compact for i # j # k # L

Let C(X,E) be the vector space of all continuous functions on a completely regular Hausdorff space X with values in a locally convex space E, equipped with the compact-open topology. In this note it will be shown that for many classes of locally convex spaces K C(X,E) lies in K if and only if C(X)=C(X,¦K) (¦K=¦R or C) and E belong to K. This is valid for the classes K. of all metrizable, normed, (DF)-, η-locally topological, separable, quasi-complete, complete, nuclear, Schwartz, semi-Montel, Montel, semi-reflexive, reflexive and quasi-normable locally convex spaces, respectively. But in general C(X,E) is not quasi-barrelled, barrelled and bornological, respectively, if C(X) and E belong to the same class. We shall give sufficient conditions for C(X,E) to be quasi-barrelled and barrelled, respectively.