Vijayabalaji and narayanan extended n normed linear space to fuzzy n normed linear space while the concept of ifnnls was introduced by vijayabalaji et al. Cho 4 introduced as a survey of the latest results on the relations between linear 2 normed spaces and normed linear spaces, completion of linear 2 normed spaces. Continuity and banach contraction principle in intuitionistic. Give an example to show that a vector space is not a normed linear space that is norm cant be defined on the vector space. The dimension of a linear space is defined as the cardinality i. A normed linear space is a vector space which also has a concept of vector length. Some classes of cesarotype difference sequences over n. In this paper, we define continuity and boundedness of linear operators in quasinormed linear space. In this paper, we define continuity and boundedness of linear operators in quasi normed linear space. Because of the triangle inequality, the function dx. The set of rational numbers q with the usual definition of addition and.
A vector space or linear space consists of the following four entities. Vijayabalaji received 24 march 2005 and in revised form 29 september 2005 the primary purpose of this paper is to introduce the notion of fuzzy nnormed linear spaceasageneralizationofnnormedspace. Note that c is also a vector space over r though a di erent one from the previous example. Lacunary iconvergence in probabilistic nnormed space. Let us see an example of incomplete normed linear space. The theory of such normed vector spaces was created at the same time as quantum mechanics the 1920s and 1930s. Metric and normed linear spaces defn a metric space is a pair x,d where x is a set and d.
Some more recent work in similar context may be found in 4, 7, 8, 15, 16. Quasinorm linear space of bounded linear operators is deduced. Ideal convergence characterization of the completion of linear nnormed spaces. Finally, we will usually simply write 0 for the neutral element 0 in a vector space. In 11 and 12 the basic properties of nprehilbert and n normed space are proved. On fixed point theorems for contraction mappings in n. A given set x may be measured by various distances in order to study the set in different ways.
The notion of an nnorm can be found in 2 and that of intuitionistic fuzzy sets in 1. In 1992, felbin 16 introduced the idea of a fuzzy norm whose associated metric is of kaleva and seikkala 20 type. Quantum physics, for example, involves hilbert space, which is a type of normed vector space with a scalar product where all cauchy sequences of vectors converge. Take any point pi and any line lj such that pi is not incident with lj. Functional analysis normed linear spaces mathematics. Why are vector spaces sometimes called linear spaces. A remark on the separability and approximation property in. In 11 and 12 the basic properties of nprehilbert and nnormed space are proved. Pdf a note on the quasinormed linear space researchgate. For the definition of dimension to be rigorous, we need two things. Since the linear space v is a commutative monoid described with additive notation, we can consider the addition of arbitrary families with.
Ideal convergence characterization of the completion of. We use this length function to setup limits for maps on normed linear spaces. I convergence in intuitionistic fuzzy nnormed linear space. Let x r 3 with usual component wise vector additions and scalar multiplications. Are they any two topological vector spaces not necessarily the same, as along as the domain is finitedimensional. On sequence spaces with elements in a sequence of real linear. Norms and metrics, normed vector spaces and metric spaces. Mathematical sciences on some nnormed sequence spaces hemen dutta 0 hamid mazaheri 1 0 department of mathematics, gauhati university, guwahati, assam 781014, india 1 department of mathematics, yazd university, yazd 89158, iran in this paper, we introduce the idea of constructing sequence spaces with elements in an nnorm space in comparison with the spaces c0, c. Every linear mapping on a finite dimensional space is.
Because of the cauchyschwarzbunyakowskyinequality, prehilbert spaces are normed spaces, and hilbert spaces are banach spaces. Tripathy and coauthors have done some significant work in this direction. The main aim of this article is to extend the notion of strongly cesaro summable and strongly lacunary summable real sequences to nnormed linear space valued nnls valued difference sequences. Consider the vector space kx of polynomials px in x with the norm p. When the space v is complete with respect to this metric, v is a banach space. Given any banach space b, there always exists a compact x such that b is equivalent to a closed linear subspace of bx satisfying. Then for a given sequence space z, we define the following sequence space. The study provided the results that helps riesz theorem be proved in probabilistic n normed spaces. X 2 0, with the properties that, for each x,y,z in x.
Pdf to text batch convert multiple files software please purchase personal license. Lacunary iconvergence in probabilistic nnormed space binod chandra tripathy a, mausumi sen b. Let v be a normed vector space for example, r2 with the euclidean norm. The idea of the limit is the same as it was in rst semester calculus. Important examples of vector spaces are the spaces kn. Since every normed linear space is a vector space but every vector space is not necessarily a normed linear space.
In this work we will give a generalization of the strong convex space, which was made for 2normed space 3. Then either there is a uniform bound m normed and banach spaces august 30, 2005 in fact, there is a dense g of such x. Vijayabalaji have extend nnormed linear space to fuzzy nnormed linear space. To prove axiom nl1 for the dual space we need to show that any point in i is incident with at least two lines. If v is a vector space over f, then 1 8 2f 0 v 0 v. Can the codomain be a different normed space and may not be finitedimensional.
R, where x and y are in e and y is a nonzero element, will be called a line. Recent examples on the web designed by architect william massie, the residence seems to be organized around a spiral metal staircase on three floors, with each level offering a linear space and scenic views framed on each end. The statespace formalism provides a framework for computing linearized models from nonlinear dierential equations, and sometimes relate the stability of the linearized model to the stability of a more complex nonlinear model. A banach space is a normed linear space that is complete. X k be a sequence of real linear nnormed spaces and w x k denote the x kvalued sequence space. The dual v of a normed vector space v is the space of all continuous linear maps from v to the base field the complexes or the reals such linear maps are called functionals. This chapter discusses the concept of a normed linear space that is a. The terms normed linear space, normed vector space, and normed space can be used interchangeably. Cho 4 introduced as a survey of the latest results on the relations between linear 2normed spaces and normed linear spaces, completion of linear 2normed spaces.
Proof since jlj 1 there is at least one line say lk. I know that i have not had enough linear algebra and exposure to higher mathematics to have a feel for why such a term is used for vector spaces and it would be great if someone could give an exposition. R r, set of real numbers is called a fuzzynnorm on xif and only if. A complete nnormed space e is called nbanach space. An algorithm is presented which will solve this problem in. We need to show that in the dual space axioms nl1 and nl2 are satis. The generalization of apollonious identity to linear nnormed. An intuitionistic fuzzy nnormed linear space or in short ifnnls is the triple x. Historia mathematica, volume 22, issue 3, 1995, pages 262303 available here from elsevier may shed some light on your question, although you may not have enough mathematical experience to understand the entire article. The primary purpose of this paper is to introduce the notion of fuzzy nnormed linear space as a generalization of nnormed space. Why should we distinguish between linear and non linear if such a term non linear space exists. Malceski 9 which are generalization of ninner product spaces introduced by a.
If the connection number for pi with lj is jljj, then s is a linear space. Defn a metric space is a pair x,d where x is a set and d. In simple words, a vector space is a space that is closed under vector addition and under scalar multiplication. It must be shown that any two points p and q are on a line. We now introduce the following sequence spaces with elements in a sequence of real linear nnormed spaces. It should be mentioned however that, although there are different ways of. We shall verify that c,jj is a normed space over both c and r, where jzj p z z. To practice dealing with complex numbers, we give the following example. Probabilistic nnormed spaces, dncompact sets and dn.
Normed linear spaces over and department of mathematics. For example, the imposing of absolute value of a variable. A systematic development of linear nnormed spaces has been extensively made by. If x is a linear space with an ideal uniformly continuous nnorm, then x.
N and x be a real linear space of dimension greater than or equal to n. Definition a banach space is a real normed linear space that is a complete metric space in. Bounded linear operators in quasinormed linear space. Hirschberg princeton university the problem of finding a longest common subse quence of two strings has been solved in quadratic time and space. Pdf in this paper, both the productnormed linear space p. On sequence spaces with elements in a sequence of real.
Vijayabalaji and narayanan extended nnormed linear space to fuzzy nnormed linear space while the concept of ifnnls was introduced by vijayabalaji et al. X, is a normed vector space or normed linear space or normed linear. I was wondering what the domain and codomain of such linear function are. Exercise 7 if v is a normed vector space, the map x.
Notes 11 1 the term vector space is very often used for what we call a linear space. Cn, with the usual operations, is a vector space over c. Then an application of the obtained results which is a well known theorem in metric spaces and shows the relations between bounded and compact sets was proved in the study, for probabilistic n normed spaces. Pdf an introduction to 2fuzzy nnormed linear spaces. In particular, every nnormed space is a normed space. Normed linear spaces over and university of nebraska. Xn are n normed linear spaces, we define a norm on the. A linear space algorithm for computing maximal common. On some nnormed linear space valued difference sequences.
Every linear function on a finitedimensional space is continuous. Denote the set of all measurable extended realvalued functions on e that are. Then, is called a 2norm and the linear space x equipped with 2norm is called a linear 2normed space. Introduce linear operators null space and range of linear operators reading for next time. Warmup exercise 3 suppose v v 1 v 2, and wis another vs. Pdf an introduction to 2fuzzy nnormed linear spaces and a. A linear nnormed space is said to be complete if every cauchy sequence is convergent to an element of e. In this work we will give a generalization of the strong convex space, which was made for 2 normed space 3. Some classes of cesarotype difference sequences over nnormed spaces hemen dutta in this paper, we introduce some cesrotype difference sequences spaces defined over a real linear nnormed space and investigate the spaces for completeness under suitable nnorm in each case. For example, ndimensional euclidean space is a normed linear. This is another example of a metric space that is not a normed vector space.
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