N ls productbanach space and productseminormed linear space productsemibanch space are introduced. On the stability of the linear functional equation, proc. In this paper, both the productnormed linear space p. Bidholi, via prem nagar, dehradun uttarakhand, india.
X 2 0, with the properties that, for each x,y,z in x. Enter your mobile number or email address below and well send you a link to download the free kindle app. Open problems in theory of metric linear spaces numdam. In particular, it is shown that for x a bounded metric space, the supremum of the mapping radii of x in all convex subsets of normed metric spaces is equal to the in. The result of mazur was a starting point for the theory of. A norm is a realvalued function defined on the vector space that has the following properties.
The geometry of metric and linear spaces proceedings of a. I wrote a book about the subject metric linear spaces first edition 1973, the second one 1985. We obtain the generalized hyersulam stability of the cauchyjensen functional equation. Isometric embedding into spaces of continuous functions. Normed vector spaces are a subset of metric spaces and a superset of inner product spaces.
Relations between absolute, unconditional and metric convergence of series are studied. Metric linear spaces mathematics and its applications hardcover june 30, 1985 by s. In many realworld applications data come as discrete metric spaces sampled around 1dimensional filamentary structures that can be seen as metric graphs. The following metrics do not arise as norms otherwise we. Download product flyer download highresolution cover. We are given a group and a metric group with metric. For all x, y, z in x, this function is required to satisfy the following conditions. Pdf in this paper, both the productnormed linear space p. Some fixed point results for generalized weak contraction. Asymptotic geometry of banach spaces and non linear quotients. Professor copsons book provides a more leisurely treatment of metric spaces than is found in books on functional analysis.
The stability problem of equations originated from a question of ulam concerning the stability of group homomorphisms. Metric linear spaces mathematics and its applications. Stability of a cauchyjensen functional equation in quasi. Click and collect from your local waterstones or get free uk delivery on orders over. Every extended metric can be transformed to a finite metric such that the metric spaces are equivalent as far as notions of topology such as continuity or convergence are concerned. Some characterizations of finitedimensional hilbert spaces. The ulam stability of functional equation on matrix quasi. State true or false giving proper justi cation for each of the following statements. On some well known fixed point theorems in bmetric spaces, turkish. A metric on a set x is a function called the distance function or simply distance.
We prove the hyersulam stability of the following jensen functional inequality in banach spaces for any fixed nonzero integer 1. Rolewicz, stefan 1987, functional analysis and control theory. We prove the existence theorem of fixed points for a generalized weak contractive mapping in modular spaces. Functional analysis and control theory linear systems. A sequence xn in a metric space e,d is said to converge to an element x of e if lim n. Linearly rigid metric spaces and kantorovich type norms. Laws of large numbers and beck convexity in metric linear spaces. In the sequel, we only consider linear spaces over the field r. We relate these questions to the general theory of norms and metrics in spaces of measures on a metric space, and introduce the notion of a banach norm compatible with a given metric. Zelazko i wrote a series of papers on linear topological. If we can measure distance, then we can reproduce many of the properties of r.
Copson, edward thomas copson, feb 11, 1988, mathematics, 152 pages. Metric linear spaces by stefan rolewicz waterstones. Later, rhoades proved that the result which alber et al. A question arises how to extend this theorem to functions defined on a metric space without any linear structure. Danuta przeworskarolewicz i stefan rolewicz im pan. Rodriguez, nonassociative normed algebras spanned by hermitian elements, proc. Differential and integral equations in banach spaces.
Rocky mountain journal of mathematics project euclid. A unification of g metric, partial metric, and b metric spaces hussain, nawab, rezaei roshan, jamal, parvaneh, vahid, and latif, abdul, abstract and applied analysis, 20. Rolewicz 44, 45 considered the class of metric vector spaces which are locally bounded and whose quasinorms are conca. Metric linear spaces mathematics monographs hardcover january 1, 1972 by stefan rolewicz author. A linear metric space with an fnorm is called f space. Definition of metric linear spaces and the theorem on the invariant norm let x be a linear space over either complex or real numbers. Find all the books, read about the author, and more. The main result is that in every complete locallybounded linear topological space there exist series which are unconditionally yet not absolutely convergent. Then you can start reading kindle books on your smartphone, tablet, or computer no kindle device required.
Journal of multivariate analysis 10, 442459 1980 laws of large numbers and beck convexity in metric linear spaces k. This book deals principally with metric linear spaces but, in many cases the notion of topological linear spaces can be a very useful tool. Woyczynski the cleveland state university communicated by s. Rolewicz, metric linear spaces, reidel and pwn, dordrecht and warszawa. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of length in the real world. In particular, it is shown that for x a bounded metric space, the supremum of the mapping radii of x in all convex subsets of normed metric spaces is equal to the in mum of the sup norms of all convex linear.
Sorry, we are unable to provide the full text but you may find it at the following locations. In 1997, alber and guerredelabriere introduced the concept of weak contraction in hilbert spaces. On the stability of a generalized quadratic and quartic type. Citeseerx document details isaac councill, lee giles, pradeep teregowda.
Throughout this paper, both e and f will denote real normed linear space. A set b is a basis for linear space x if every element in x can be written uniquely as a. On the stability of a generalized quadratic and quartic. Metric linear spaces mathematics monographs hardcover january 1, 1972 by stefan rolewicz author visit amazons stefan rolewicz page. In mathematics, a normed vector space is a vector space on which a norm is defined. Buy metric linear spaces by stefan rolewicz from waterstones today.
Given a set b in a linear space, the span of b, denoted spanb, is the set of all. Limit properties of ordered families of linear metric spaces z. Let f be a realvalued convex continuous function defined on an open convex subset. Rolewicz, metric linear spaces, instytut matematyczny polskiej akademi nauk.
The most significant class of quasibanach spaces, which are not banach spaces are lp spaces for 0 p 1 equipped with the lp norms kkp. Note that condition 1 and 2 together produce positive. Systems governed by ordinary differential equations. Oct 07, 2010 a walk through of linear spaces vector spaces metric spaces normed linear spaces inner product spaces euclidean spaces function spaces functionals. Furthermore, the most complete development of these theories is due to mazur, luxemburg, and turpin 7 9. We prove the hyersulam stability of the following jensen functional inequality in banach spaces for any fixed nonzero integer. Of the many variations of vector measures, the frechet variation is finite valued but only subadditive. In this paper we address the metric reconstruction problem of such filamentary structures from data sampled around them. N ls product banach space and productseminormed linear space. Then d is a metric on e called the uniform metric or uniform distance function. A linear space x is called a topological linear space if it is a hausdorff space and if the operation of addition of elements and the operation of multiplication by scalars are continuous. Lectures on some fixed point theorems of functional analysis.
This can be done using a subadditive monotically increasing bounded function which is zero at zero, e. In the present paper we investigate the general solution of the functional equation 1. Tools are developed for estimating the mapping radius of a metric space x in a metric space y. In mathematics, a normed vector space is a vector space over the real or complex numbers, on which a norm is defined. Laws of large numbers and beck convexity in metric linear. Download here metric linear spaces, stefan rolewicz, 1985, mathematics, 458 pages. Shrinkable neighborhoods in hausdorff linear spaces, math. On series in linear topological spaces springerlink. A walk through of linear spaces vector spaces metric spaces normed linear spaces inner product spaces euclidean spaces function spaces functionals. Abstract in this paper, we define the matrix quasinormed spaces and prove the ulam stability of functional equation deriving from cubic and quadratic functions 6f. Let x be a metric linear space x with metric px, y, i. Asymptotic geometry of banach spaces and non linear. It is easy to see that the function is a solution of the functional equation 1.
The main part of our considerations will be the same in both cases. Przeworska rolewicz i was working in the algebraic approach to the theory of linear operators and we wrote a common book equations in linear spaces 1968. Hyersulam stability of jensen functional inequality in p. Functional analysis and control theory linear systems stefan. The unit ball of is the closed convex hull of its isometries. By a quasibanach space we mean a complete quasinormed space, i. Recall that the theory of modular on linear spaces and the corresponding theory of modular linear spaces were founded by nakano 4, 5 and redefined by musielak and orlicz. Examples of metric spaces which are not normed linear spaces.
Metric linear spaces mathematics and its applications hardcover june 30, 1985. The unit ball of, the space of linear and continuous operators in, is the convex hull of its isometries. A normed linear space is a linear space endowed with a norm. Lipschitz properties of convex functions cobzas, stefan, advances in operator theory, 2017. We prove that they can be approximated, with respect to the gromovhausdorff distance. We prove that each of the above are metric spaces by showing that they are normed linear spaces, where the obvious candidates are used for norms. Recently, lima and randrianarivony pointed out the role of the property.