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A unified approach, that uses countable sets with convergences and basics of the theory of LF-spaces, is given to some old results of the Silverman–Toeplitz type obtained in the late 1930's by H.J. Hamilton for matrix transformations of multiple sequences.

In this paper, we present two new fixed point theorems in Fr?chet algebras and Fr?chet spaces. Our fixed point results are expressed with the help of family of measures of noncompactness and generalizes Darbo theorem. As an application, we establish some existence results for various types of nonlinear integral equations.

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.

Suppose M is a non-compact connected n -manifold without boundary, D ( M ) is the group of C ∞ -diffeomorphisms of M endowed with the Whitney C ∞ -topology and D 0 ( M ) is the identity connected component of D ( M ) , which is an open subgroup in the group D c ( M ) ⊂ D ( M ) of compactly supported diffeomorphisms of M . It is shown that D 0 ( M ) is homeomorphic to N × R ∞ for an l 2 -manifold N whose topological type is uniquely determined by the homotopy type of D 0 ( M ) . For instance, D 0 ( M ) is homeomorphic to l 2 × R ∞ if n = 1 , 2 or n = 3 and M is orientable and irreducible. We also show that for any compact connected n -manifold N with non-empty boundary ∂ N the group D 0 ( N ∖ ∂ N ) is homeomorphic to D 0 ( N ; ∂ N ) × R ∞ , where D 0 ( N ; ∂ N ) is the identity component of the group D ( N ; ∂ N ) of diffeomorphisms of N that do not move points of the boundary ∂ N .

The weak and weak* topologies are introduced for Banach spaces and their duals, and the Banach-Alaoglu theorem is proved. Further topologies are introduced on the spaces of bounded linear operators, and these are placed in the context of the general locally convex vector spaces. Properties of convex bodies are discussed, including the Krein-Milman and Choquet theorems, and several applications of weak* compactness are given. These include Furstenberg’s proof of Weyl’s polynomial equidistribution theorem and elliptic regularity for the Laplace operator at the boundary. The theory of distributions is introduced.

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.

Let X be a locally compact Polish space and G a non-discrete Polish ANR group. By C(X,G), we denote the topological group of all continuous maps f:X \to G endowed with the Whitney (graph) topology and by C_c(X,G) the subgroup consisting of all maps with compact support. It is known that if X is compact and non-discrete then the space C(X,G) is an l_2-manifold. In this article we show that if X is non-compact and not end-discrete then C_c(X,G) is an (R^\infty \times l_2)-manifold, and moreover the pair (C(X,G), C_c(X,G)) is locally homeomorphic to the pair of the box and the small box powers of l_2., 11 pages

In this paper n denotes a natural number, X denotes a set, and F1, G1 denote families of subsets of X. Let us consider X, F1. We say that F1 is finite-order if and only if: (Def. 1) There exists n such that for every G1 such that G1 ⊆ F1 and n ∈ CardG1 holds ⋂ G1 is empty. Let us consider X. Observe that there exists a family of subsets of X which is finite-order and every family of subsets of X which is finite is also finite-order. Let us considerX, F1. The functor orderF1 yielding an extended real number is defined as follows: (Def. 2)(i) For everyG1 such that orderF1+1 ∈ CardG1 andG1 ⊆ F1 holds ⋂ G1 is empty and there exists G1 such that G1 ⊆ F1 but CardG1 = orderF1+1 but ⋂ G1 is non empty or G1 is empty if F1 is finite-order, (ii) orderF1 = +∞, otherwise.

In this paper, we prove that an L-topological vector space is locally convex (locally bounded) iff its induced I(L)-topological vector space is locally convex (locally bounded), where I(L) is the L-fuzzy unit interval. Also, we give a necessary and sufficient condition for the induced I(L)-topological vector space to be I(L)-fuzzy normable.

In this paper, the relation between two definitions of locally convex I -topological vector spaces is studied. These two definitions are introduced by Katsaras [Fuzzy topological vector spaces II, Fuzzy Sets and Systems 12 (1984) 143–154] and Wu and Li [Convexity and fuzzy topological vector spaces, Science Exploration (China) 4(1) (1984) 1–4], respectively. We prove that the locally convex I -topological vector space in the sense of Wu and Li is a special case of that in the sense of Katsaras. We introduce the notion of generalized fuzzy semi-norm and prove that every locally convex I -vector topology in the sense of Katsaras on X can be determined by a family of generalized fuzzy semi-norms on X .

Weak and strong continuity conditions for representations of topological groups in locally convex spaces are considered. In particular, weak continuity conditions of reducible locally equicontinuous representations of a topological group in a locally convex space that define weakly continuous representations in an invariant subspace and in the quotient space by this invariant subspace are investigated. These conditions help one to prove the weak continuity of averages and approximations related to weakly continuous locally equicontinuous quasirepresentations of amenable topological groups. Strong continuity conditions for a representation approximating a quasirepresentation of this kind are related to conditions of automatic strong continuity of weakly continuous representations, and fail to hold for some groups, spaces, and representations. In this connection, strong continuity conditions for weakly continuous representations in quasicomplete barrelled locally convex spaces are indicated for a broad class of topologically pseudocomplete groups (which includes the Cech complete groups and the locally pseudocompact groups). Several examples are discussed, in particular, ones relating to the construction of -products with distinguished subgroups.

For all linear and many nonlinear operators between Frechet spaces, several continuity results are established. Especially, a new closed graph theorem is given.

In this article, we present a method to obtain new Ekeland-type variational principles for mappings defined between locally convex spaces. The method is based on a special nuclear cone defined in a product of two locally convex spaces and on Pareto efficiency with respect to this cone.

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 $X$.

In this paper, using the theory of L-topological vector spaces, we study some properties of inductive topologies determined by the family of L-fuzzy linear order-homomorphisms defined by Fang Jin-xuan (J. Fuzzy Math. 4(1) (1996) 93), and give a characterization of inductive topologies determined by a single L-fuzzy linear order-homomorphism. As an example, we point out the quotient L-topology of L-tvs is an inductive topology.

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.

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.

We shall construct in ZFC two Frechet–Urysohn α 4 -spaces, the product of which is α 4 , but fails to be Frechet–Urysohn. This answers Nogura's question from 1985.

We prove that for a metrizable space X the following are equivalent: (i) the free Abelian topological group A(X) is the inductive limit of the sequence {A n (X):n∈ N } , where A n (X) is formed by all words of reduced length ≤n ; (ii) X is locally compact and the set of all non-isolated points of X is separable. In the non-Abelian case, for a metrizable X the following are equivalent: (i) the free topological group F(X) is the inductive limit of the sequence {F n (X):n∈ N } ; (ii) X is either locally compact separable or discrete.

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.

The notion of bounded element of C*-inductive locally convex spaces (or C*-inductive partial *-algebras) is introduced and discussed in two ways: The first one takes into account the inductive structure provided by certain families of C*-algebras; the second one is linked to the natural order of these spaces. A particular attention is devoted to the relevant instance provided by the space of continuous linear maps acting in a rigged Hilbert space.

Countable spaces with an only nonisolated point for which the free topological groups are the inductive limits of the sets of words having restricted length are characterized. It is proved for a collectionwise normal X that if its free topological group is the inductive limit of the sets of words having restricted length, then any closed discrete set of non-P-points in X is countable. All the results obtained are valid for both Abelian and non-Abelian free groups.

A strictly barrelled diskBin a Hausdorff locally convex spaceEis a disk such that the linear span ofBwith the topology of the Minkowski functional ofBis a strictly barrelled space. Valdivia's closed graph theorems are used to show that closed strictly barrelled disk in a quasi-(LB)-space is bounded. It is shown that a locally strictly barrelled quasi-(LB)-space is locally complete. Also, we show that a regular inductive limit of quasi-(LB)-spaces is locally complete if and only if each closed bounded disk is a strictly barrelled disk in one of the constituents.

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.

In this paper, we propose a novel compressive sensing (CS) based indoor positioning approach, which uses the signal strength differentials (SSDs) as location fingerprints (LFs). The target location is regarded as an unknown sparse location vector in the discrete spatial domain. Then it just takes a little number of online noisy SSD measurements for the exact recovery of the sparse location vector by solving an l 1 -minimization program. In order to mitigate the influences of large measurements noise on the recovery accuracy, an LF space denosing algorithm is proposed to discriminate the localization contribution rate of every LF according to its SSD variation. Moreover, an LF space filtering strategy is also exploited to lower the high computational complexity of the CS recovery algorithm. Both experimental results and simulations demonstrate that we achieve remarkable improvements on the positioning performance of the CS based approach by using the two proposed algorithms.

For an order-continuous Banach function space Λ and a separated inductive limit E:= indnEn, we prove that indn A {En} is a topological subspace of Λ {E}; moreover, both spaces coincide if the inductive limit is hyperstrict. As a consequence, we deduce that if E is an LF-space, then Lp {E} is barrelled for 1 ≤ p ≤ ∞.

In this paper we give a characterization of p-adic semi-Montel spaces which allows us to describe the finest polar semi-Montel topology coarser than the original topology of a locally convex space. As an application we derive that every polar semi-Montel space is a polar inductive limit of a family of nuclear spaces. We also pay attention to the connection with compactifying operators.