Fréchet Spaces via Fuzzy Structures

The important theme of this paper is to define a lower and upper approximation for the set under consideration in order to examine a novel class of topological field known as Nano fuzzy soft topological space. Fuzzy topology coverts ordered structure to topological structure by using by its own criteria conditions. Nano topology is defined with the help of equivalence relation so that even micro spacing is included in the topological space. The condition of these two topological space has been merged and a new space known as Nano fuzzy topological space has been investigated and studied. Soft sets are an abstraction of fuzzy set theory which was interposed to deal with uncertainty conditions using parameters. Using soft sets, Nano fuzzy topology can be extended to a new form called Nano fuzzy soft topological space and it can be used to study matters of vagueness by defining a proper boundary layer. Since this concept depends upon defining appropriate boundary layers using topological concepts, it easy to determine the key components for a particular problem by properly defining three important regions which are upper approximation area, lower approximation area and boundary region. The use of Nano fuzzy soft topological space in predicting the cause of a disease has been discussed in this paper. It can be extended to study any complex problems in real life by converting it to a Nano fuzzy soft topological space using the three main key factors.

Let (X, =) be a topological space. A subset A of X is called pre-open if A ⊆ Int(Cl(A)). Let P O(X) denote the family of all pre-open sets in X. In general, P O(X) does not form a topology on X. Furthermore, in topological vector spaces, it is not always true that P O(L) forms a topology on L where L is a topological vector space. In this note, we prove that the class of strongly preirresolute topological vector spaces is that subclass of topological vector spaces in which P O(L) forms a topology and thereby we see that all proved results in [5] concerning strongly preirresolute topological vector spaces are obvious.

The notion of fuzzy sets initiated to overcome the uncertainty of an object. Fuzzy topological space, in- tuitionistic fuzzy sets in topological structure space, vagueness in topological structure space, rough sets in topological space, theory of hesitancy and neutrosophic topological space, etc. are the extension of fuzzy sets. Soft set is a family of parameters which is also a set. Fuzzy soft topological space, intuitionistic fuzzy soft and neutrosophic soft topological space are obtained by incorporating soft sets with various topological structures. This motivates to write a review and study on various soft set concepts. This paper shows the detailed review of soft topological spaces in various sets like fuzzy, Intuitionistic fuzzy set and neutrosophy. Eventually, we compared some of the existing tools in the literature for easy understanding and exhibited their advantages and limitations.

A note on multiplier convergent series

This chapter pursues three lines of investigation in the subject of functional analysis—one involving smooth functions and distributions, one involving fixed-point theorems, and one involving spectral theory.Section 1 introduces topological vector spaces. These are real or complex vector spaces with a Hausdorff topology in which addition and scalar multiplication are continuous. Examples include normed linear spaces, spaces given by a separating family of countably many seminorms, and weak and weak-star topologies in the context of Banach spaces. Various general properties of topological vector spaces are proved, and it is proved that the quotient of a topological vector space by a closed vector subspace is Hausdorff and is therefore a topological vector space.Section 2 introduces a topology on the space C∞(U) of smooth functions on an open subset of ℝN. The support of a continuous linear functional on C∞(U) is defined and shown to be a compact subset of U. Accordingly, the continuous linear functionals are called distributions of compact support.Section 3 studies weak and weak-star topologies in more detail. The main result is Alaoglu’s Theorem, which says that the closed unit ball in the weak-star topology on the dual of a normed linear space is compact. In an earlier chapter a preliminary form of this theorem was used to construct elements in a dual space as limits of weak-star convergent subsequences.Section 4 follows Alaoglu’s Theorem along a particular path, giving what amounts to a first example of the Gelfand theory of Banach algebras. The relevant theorem, known as the Stone Representation Theorem, says that conjugate-closed uniformly closed subalgebras containing the constants in B(S) are isomorphic via a norm-preserving algebra isomorphism to the space of all continuous functions on some compact Hausdorff space. The compact space in question is the space of multiplicative linear functionals on the subalgebra, and the proof of compactness uses Alaoglu’s Theorem.Sections 5–6 return to the lines of study toward distributions and fixed-point theorems. Section 5 studies the relationship between convexity and the existence of separating linear functionals. The main theorem makes use of the Hahn-Banach Theorem. Section 6 introduces locally convex topological vector spaces. Application of the basic separation theorem from the previous section shows the existence of many continuous linear functionals on such a space.Section 7 specializes to the line of study via smooth functions and distributions. The topic is the introduction of a certain locally convex topology on the space C ∞com (U) of smooth functions of compact support on U. This is best characterized by a universal mapping property introduced in the section.Sections 8–9 pursue locally convex spaces along the other line of study that split off in Section 5. Section 8 gives the Krein-Milman Theorem, which asserts the existence of a supply of extreme points for any nonempty compact convex set in a locally convex topological vector space. Section 9 relates compact convex sets to the subject of fixed-point theorems.Section 10 takes up the abstract theory of Banach algebras, with particular attention to commutative C* algebras with identity. Three examples are the algebras characterized by the Stone Representation Theorem, any L∞ space, and any adjoint-closed commutative Banach algebra consisting of bounded linear operators on a Hilbert space and containing the identity.Section 11 continues the investigation of the last of the examples in the previous section and derives the Spectral Theorem for bounded self-adjoint operators and certain related families of operators. Powerful applications follow from a functional calculus implied by the Spectral Theorem. The section concludes with remarks about the Spectral Theorem for unbounded self-adjoint operators.KeywordsBanach AlgebraTopological Vector SpaceNormed Linear SpaceInternal PointCompact Hausdorff SpaceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The concept of complement function is used to define a fuzzy closed subset of a fuzzy topological space. That is a fuzzy subset l is fuzzy closed if the standard complement 1-l = l¢ is fuzzy open. Here the standard complement is obtained by using the function C: [0, 1]® [0, 1] defined by C (x) = 1-x, for all x I[0, 1]. Several fuzzy topologists used this type of complement while extending the concepts in general topological spaces to fuzzy topological spaces. But there are other complements in the fuzzy literature. This motivated the second and third authors to introduce the concepts of fuzzy C -closed sets, and fuzzy C - pre closed sets, in fuzzy topological spaces. In this paper, we generalize the concept of fuzzy strongly pre open and fuzzy strongly pre closed sets by using the arbitrary complement function C, instead of the usual fuzzy complement function, and by using fuzzy C – pre closure instead of fuzzy pre closure. The concepts of fuzzy strongly C – pre open and strongly C – pre closed sets in fuzzy topological spaces and studied their properties in the third and fourth section, respectively. Key words: Fuzzy strongly C – pre open, fuzzy strongly C – pre closed sets and fuzzy topology.

On locally convex I-topological vector spaces

The main objective of this paper is to introduce a new generalized form of topological vector spaces, namely, almost $\\alpha$-topological vector spaces by using the concepts of $\\alpha$-open sets, regular open sets, and almost $\\alpha$-continuous mappings. In the present paper, some important examples of almost $\\alpha$-topological vector spaces with basic properties are characterized.

The spaces of test functions we are going to use are vector spaces of smooth (i.e., sufficiently often continuously differentiable) functions on open nonempty subsets Ω ⊆ ℝ n equipped with a ‘natural’ topology. Accordingly we start with a general method to equip a vector space V with a topology such that the vector space operations of addition and scalar multiplication become continuous, i.e., such that $$ \begin{array}{*{20}c} {_{M\,:\,\,\mathbb{K}\, \times \,V\, \to \,V,}^{A\,:\,\,V\, \times \,V\, \to \,V,} } & {_{M(\lambda ,\,x)\, = \,\lambda x,\,}^{A(x,\,y)\, = \,x\, + \,y,} } & {_{\lambda \, \in \,\mathbb{K},\,\,\,x\, \in \,V}^{x,\,y\, \in \,V} } \end{array} $$ become continuous functions for this topology. This can be done in several different but equivalent ways. The way we describe has the advantage of being the most natural one for the spaces of test functions we want to construct. A vector space V which is equipped with a topology T such that the functions A} and M are continuous is called a topological vector space, usually abbreviated as TVS. The test function spaces used in distribution theory are concrete examples of topological vector spaces where, however, the topology has the additional property that every point has a neighborhood basis consisting of (absolutely) convex sets. These are called locally convex topological vector spaces, abbreviated as LCVTVS.

Epileptic seizures are manifestations of epilepsy caused a temporary electrical disturbance in a group of brain cells. Electroencephalography (EEG) is a recording of electrical activity of the brain and it contains valuable information related to the different physiological states of the brain. Flat EEG (fEEG) was developed to compress and analyze information in the brain during epileptic seizures obtained from EEG signal. In this study, the fEEG is made in digital form by using Voronoi digitization. Khalimsky fuzzy topology and fuzzy topological digital space is constructed to be used in studying properties fuzzy topology of fEEG. It is obtained that fEEG is a fuzzy topological digital space and τ-connected.

Let K be a compact Hausdorff space and let )~ be a probability measure on K. We denote by Lo(K,2) the space of all Borel functions f:K--*N with the topology of convergence in measure. Lo(K, 2) is an F-space (complete metric topological vector space) if, as usual, we identify functions equal almost everywhere. Spaces of the type Lo(K,2 ) are probably the most studied examples of non-locally convex topological vector spaces. In spite of their bad reputation there is some evidence that they are in fact rather well-behaved spaces. Thus for a general topological vector space X it is often very useful to be able to produce nontrivial linear operators T:X--*Lo(K,2 ), in the same way as linear functionals facilitate the theory of locally convex spaces. We shall say that a point xeX is pathological it wherever T:X-*Lo(K, 2) is a continuous linear operator then Tx=O. X is pathological if every xsX is pathological. Note that if X is separable then it suffices in the definition to take K =(0, 1) with Lebesgue measure. The first example of a pathological F-space was given in 1973 by Christensen and Herer [3]. A more natural example is the space Lp/Hp where 0 < p < 1 (see [1, 5]). However the Christensen-Herer example also showed the connection with pathological submeasures. We recall that if d is an algebra of subsets of some abstract set L then a submeasure 4~:d--,lR is a map satisfying ~b(A)<_~b(AwB)<d~(A)+O(B) and th(O) = 0. 4~ is said to be pathological (see [3, 11]) if whenever 2 : d ~ N is a (finitelyadditive) measure with 0 <)~(A) < th(A) (At d ) then )~0. Now if S(~) denotes the space of all simple d-measurable functions f:L~IR, then S(~) can be topologized by the topology of convergence in @measure (functions differing only on a set of @measure zero are identified). The completion of S(~) in this topology may be denoted by A(~b). Then A(q~) is pathological

If A and B are locally convex topological vector spaces, and B has certain additional structure, then the space L(A, B) of all continuous linear mappings of A into B is characterized, within isomorphism, as the inductive limit of a family of spaces, whose elements are functions, or measures. The isomorphism is topological if L(A, B) is given a particular topology, defined in terms of the seminorms which define the topologies of A and B. The additional structure on B enables L(A, B) to be constructed, using the duals of the normed spaces obtained by giving A the topology of each of its seminorms separately.

In this paper, the concept of particle swarm optimization (PSO) with support vector machine (SVM) in fuzzy topological space is proposed for classifying a given image. SVMs have their roots in statistical learning theory and have gained prominence because they are more accurate and reliable even if a limited number of training samples are used. However, the task of parameter selection in SVMs is quite tedious, as they require the selection of the kernel function parameter. In order to overcome this problem, the support vector machine is trained using the particle swarm optimization technique. In the proposed approach, this classification scheme is applied to the content selected for the classification. Moreover, in order to deal with the mixed pixel problem the input image class in the spectral space is decomposed into three parts: the interior, the boundary and the exterior in fuzzy topological space (fts). This decomposition is done using a threshold value based on the optimal intercorrelation coefficient. The interior-class pixels are classified using the PSO-SVM. The boundary-class pixels that are the fuzzy ones are reclassified on the basis of the connectivity theory of the fuzzy topology. The results of the experiment conducted on the remotely sensed image indicate that the proposed approach is more accurate than the existing fuzzy topology integrated support vector machine (FTSVM). Thus, it provides a better classification scheme.

In this paper, we introduce the notion of almost \(s-\)topological vector spaces and present some examples and counterexamples of almost \(s-\)topological vector spaces. Some general properties of almost \(s-\)topological vector spaces are investigated.

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).