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Some new bounds for Cebysev functional for sequences of vectors in normed linear spaces are pointed out.

Based on ideas of R.W. Cross, a simplified proof is presented of the density invariance of certain operational quantities associated with bounded linear operators in normed vector spaces.

Let C be a nonempty closed convex subset of a Banach space $E$. In this paper we introduce a new definition for variational inequality V I (C, B) on E that generalizes the analogue definition on Hilbert spaces. We generalize (u, v)-cocoercive mappings and v-strongly monotone mappings from Hilbert spaces to Banach spaces. Then we prove the generalized variational inequality V I (C, B) is singleton for (u, v)-cocoercive mappings under appropriate assumptions on Banach spaces that extends and improves [S. Saeidi, Comments on relaxed (u, v)-cocoercive mappings. Int. J. Nonlinear Anal. Appl. 1 (2010) No. 1, 54-57].

We introduce the notion of strongly orthogonal set relative to an element in the sense of Birkhoff-James in a normed linear space to find a necessary and sufficient condition for an element $ x $ of the unit sphere $ S_{X}$ to be an exposed point of the unit ball $ B_X .$ We then prove that a normed linear space is strictly convex iff for each element x of the unit sphere there exists a bounded linear operator A on X which attains its norm only at the points of the form $ \lambda x $ with $ \lambda \in S_{K} $.

Let $B_Y$ denote the unit ball of a normed linear space $Y$. A symmetric, bounded, closed, convex set $A$ in a finite dimensional normed linear space $X$ is called a {\it sufficient enlargement} for $X$ if, for an arbitrary isometric embedding of $X$ into a Banach space $Y$, there exists a linear projection $P:Y\to X$ such that $P(B_Y)\subset A$. The main results of the paper: {\bf (1)} Each minimal-volume sufficient enlargement is linearly equivalent to a zonotope spanned by multiples of columns of a totally unimodular matrix. {\bf (2)} If a finite dimensional normed linear space has a minimal-volume sufficient enlargement which is not a parallelepiped, then it contains a two-dimensional subspace whose unit ball is linearly equivalent to a regular hexagon.

Carry out �standard� constructions in (linear) functional analysis: Abstract Hilbert space � one in each dimension, Concrete Hilbert space � Many, such as L2([0, 1])

Carry out �standard� constructions in (linear) functional analysis: Abstract Hilbert space � one in each dimension, Concrete Hilbert space � Many, such as L2([0, 1])

It is well known that every (real or complex) normed linear space $L$ is isometrically embeddable into $C(X)$ for some compact Hausdorff space $X$. Here $X$ is the closed unit ball of $L^*$ (the set of all continuous scalar-valued linear mappings on $L$) endowed with the weak$^*$ topology, which is compact by the Banach-Alaoglu theorem. We prove that the compact Hausdorff space $X$ can indeed be chosen to be the Stone-Cech compactification of $L^*\setminus\{0\}$, where $L^*\setminus\{0\}$ is endowed with the supremum norm topology.

Book Source: Digital Library of India Item 2015.194108 dc.contributor.author: Saran Behari Mathur dc.date.accessioned: 2015-07-08T03:56:59Z dc.date.available: 2015-07-08T03:56:59Z dc.date.digitalpublicationdate: 2005-08-30 dc.identifier.barcode: 1990010092372 dc.identifier.origpath: /rawdataupload/upload/0092/372 dc.identifier.copyno: 1 dc.identifier.uri: http://www.new.dli.ernet.in/handle/2015/194108 dc.description.scannerno: 5 dc.description.scanningcentre: IIIT, Allahabad dc.description.main: 1 dc.description.tagged: 0 dc.description.totalpages: 113 dc.format.mimetype: application/pdf dc.language.iso: English dc.publisher: Indian Institute Of Technology Kanpur dc.rights: In Public Domain dc.source.library: Indian Institute Of Technology Kanpur dc.subject.classification: Natural Sciences dc.subject.classification: Mathematics dc.subject.classification: Geometry dc.title: Some Results On Tringle Contractive Maps And Orthogonality In Normed Linear Spaces

We study the properties of rectangular constant $ \mu(\mathbb{X}) $ in a normed linear space $\mathbb{X}$. We prove that $ \mu(\mathbb{X}) = 3$ iff the unit sphere contains a straight line segment of length 2. In fact, we prove that the rectangular modulus attains its upper bound iff the unit sphere contains a straight line segment of length 2. We prove that if the dimension of the space $\mathbb{X}$ is finite then $\mu(\mathbb{X})$ is attained. We also prove that a normed linear space is an inner product space iff we have sup$\{\frac{1+|t|}{\|y+tx\|}$: $x,y \in S_{\mathbb{X}}$ with $x\bot_By\} \leq \sqrt{2}$ $\forall t$ satisfying $|t|\in (3-2\sqrt{2},\sqrt{2}+1)$.

Some sharp discrete inequalities in normed linear spaces are obtained. New reverses of the generalised triangle inequality are also given.

Steepest descent and related topics are discussed. An example is given taken from a simple rendezvous problem in control theory. The problem is one of minimizing a norm on an affine subspace. The problem here is solved in the primal.

In this paper, we shall investigate and analyse a new study on the best simultaneous approximation in the context of linear 2-normed spaces inspired by Elumalai and his coworkers in Elumalai. The basis of this investigation is to extend and refinement the definition of the classical aproximation, best approximation and some related concepts to linear 2-normed spaces.

Some sharp inequalities of Gruss type for sequences of vectors in real or complex normed linear spaces are obtained. Applications for the discrete Fourier and Mellin transform are given. Estimates for polynomials with coefficients in normed spaces are provided as well.

Let E be a normed linear space, A a bounded set in E, and G in an arbitrary set in E. The relative Chebyshev center of A in G is the set of points in G best approximating A. We have obtained elsewhere general results characterizing the spaces in which the center reduces to a singleton in terms of structural properties related to uniform and strict convexity. In this paper an analysis of the Chebyshev norm case, which falls outside the scope of the previous analysis, is presented. (Author)

In this paper, we introduce the notions of deferred statistically convergent sequence and deferred statistically Cauchy sequence in a fuzzy normed linear space(FNS) and establish some basic facts. We also define deferred statistical limit points and deferred statistical cluster points of a sequence in FNS and investigate the relations between these concepts.

In this paper, we introduce cone normed linear space, study the cone convergence with respect to cone norm. Finally, we prove the completeness of a finite dimensional cone normed linear space.

Definition. A symmetric with respect to 0 bounded closed convex set A in a finite dimensional normed space X is called a sufficient enlargement for X (or of B(X)) if for arbitrary isometric embedding of X into a Banach space Y there exists a projection P:Y\to X such that P(B(Y)) is a subset of A (by B(X) we denote the unit ball). The notion of sufficient enlargement is implicit in the paper: B.Grunbaum, Projection constants, Trans. Amer. Math. Soc. 95 (1960) 451--465. It was explicilty introduced by the author in: M.I.Ostrovskii, Generalization of projection constants: sufficient enlargements, Extracta Math., 11 (1996), 466--474. The main purpose of the present paper is to continue investigation of sufficient enlargements started in the papers cited above. In particular the author investigate sufficient enlargements whose support functions are in some directions close to those of the unit ball of the space, sufficient enlargements of minimal volume, sufficient enlargements for euclidean spaces.

‎In this paper‎, ‎we use the definition of fuzzy normed spaces given by Bag and Samanta and provide four types of fuzzy versions of contraction‎. ‎We show that these mappings necessarily have unique fixed points in fuzzy normed linear spaces‎. ‎We will show that the presented theorems are indeed fuzzy extensions of their classical counterparts‎.

Intuitionistic Fuzzy derivative, Intuitionistic Fuzzy Gateaux derivative, Intuitionistic Fuzzy Fr\'{e}chet derivative are defined and a few of their properties are studied. The relation between Intuitionistic Fuzzy Gateaux derivative and Intuitionistic Fuzzy Fr\'{e}chet derivative are emphasized.

Intuitionistic Fuzzy derivative, Intuitionistic Fuzzy Gateaux derivative, Intuitionistic Fuzzy Fr\'{e}chet derivative are defined and a few of their properties are studied. The relation between Intuitionistic Fuzzy Gateaux derivative and Intuitionistic Fuzzy Fr\'{e}chet derivative are emphasized.

In this paper, we show that every classical inner product on a linear space induces the fuzzy inner product and fuzzy norm in the sense of Bag and Samanta. We prove the Cauchy-Schwarz inequality on fuzzy Hilbert spaces. We also define fuzzy frame on fuzzy Hilbert spaces. As is known, C∞(Ω) is not normable in classical Hilbert space, but in this paper we show that C∞(Ω) is normable in fuzzy Hilbert space and so defining fuzzy frame on C∞(Ω) is possible.