scholarly journals On orthonormal sets in inner product quasilinear spaces

2016 ◽  
Vol 25 (2) ◽  
pp. 237-247
Author(s):  
YILMAZ YILMAZ ◽  
◽  
HACER BOZKURT ◽  
SUMEYYE CAKAN ◽  
◽  
...  
Keyword(s):  
Orthonormal Basis ◽  
Steklov Inst ◽  
Schauder Basis ◽  
Inner Product ◽  
Multivalued Mappings ◽  
Linear Spaces ◽  
Orthonormal Sequence ◽  
Complete Set

Aseev, S. M [Aseev, S. M., Quasilinear operators and their application in the theory of multivalued mappings, Proc. Steklov Inst. Math., 2 (1986), 23–52] generalized linear spaces by introducing the notion of quasilinear spaces in 1986. Then, special quasilinear spaces which are called ”solid floored quasilinear spaces” were defined and their some properties examined in [C¸ akan, S., Some New Results Related to Theory of Normed Quasilinear Spaces, Ph.D. Thesis, ˙Inon¨ u University, Malatya, 2016]. In fact, this classification was made so as to examine consistent and detailed some properties related ¨ to quasilinear spaces. In this paper, we present some properties of orthogonal and orthonormal sets on inner product quasilinear spaces. At the same time, the mentioned classification is crucial for define some topics such as Schauder basis, complete orthonormal sequence, orthonormal basis and complete set and some related theorems. Also, we try to explain some geometric differences of inner product quasilinear spaces from the inner product (linear) spaces.

The Contraction Principle for Multivalued Mappings on a Modular Metric Space with a Graph

2016 ◽  
Vol 59 (01) ◽  
pp. 3-12 ◽  
Author(s):  
Monther Rashed Alfuraidan
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Metric Space ◽  
Fixed Point Theorems ◽  
Metric Spaces ◽  
Orlicz Spaces ◽  
Multivalued Mappings ◽  
Linear Spaces ◽  
Modular Metric Spaces ◽  
Modular Spaces

Abstract We study the existence of fixed points for contraction multivalued mappings in modular metric spaces endowed with a graph. The notion of a modular metric on an arbitrary set and the corresponding modular spaces, generalizing classical modulars over linear spaces like Orlicz spaces, were recently introduced. This paper can be seen as a generalization of Nadler and Edelstein’s fixed point theorems to modular metric spaces endowed with a graph.

Eigenvalues of the Curvature Operator for Certain Homogeneous Manifolds

1990 ◽  
Vol 42 (6) ◽  
pp. 981-999
Author(s):  
J. E. D'Atri ◽  
I. Dotti Miatello
Keyword(s):  
Riemannian Manifold ◽  
Tangent Space ◽  
Orthonormal Basis ◽  
Riemann Tensor ◽  
Inner Product ◽  
Curvature Operator ◽  
Curvature Function ◽  
Homogeneous Manifolds ◽  
Exterior Power ◽  
Image Position

Given a Riemannian manifold M, the Riemann tensor R induces the curvature operator on the exterior power of the tangent space, defined by the formula where the inner product is defined by From the symmetries of R, it follows that ρ is self-adjoint and so has only real eigenvalues. R also induces the sectional curvature function K on 2-planes in is an orthonormal basis of the 2-plane π.

SHAPIRO’S UNCERTAINTY PRINCIPLE IN THE DUNKL SETTING

2015 ◽  
Vol 92 (1) ◽  
pp. 98-110 ◽  
Author(s):  
SAIFALLAH GHOBBER
Keyword(s):  
Uncertainty Principle ◽  
Orthonormal Basis ◽  
Integral Transform ◽  
Reflection Group ◽  
Dunkl Transform ◽  
Time Frequency ◽  
Orthonormal Bases ◽  
Orthonormal Sequence ◽  
Finite Reflection Group ◽  
Uniformly Bounded

The Dunkl transform ${\mathcal{F}}_{k}$ is a generalisation of the usual Fourier transform to an integral transform invariant under a finite reflection group. The goal of this paper is to prove a strong uncertainty principle for orthonormal bases in the Dunkl setting which states that the product of generalised dispersions cannot be bounded for an orthonormal basis. Moreover, we obtain a quantitative version of Shapiro’s uncertainty principle on the time–frequency concentration of orthonormal sequences and show, in particular, that if the elements of an orthonormal sequence and their Dunkl transforms have uniformly bounded dispersions then the sequence is finite.

scholarly journals On normal derivations of Hilbert–Schmidt type

1987 ◽  
Vol 29 (2) ◽  
pp. 245-248 ◽  
Author(s):  
Fuad Kittaneh
Keyword(s):  
Hilbert Space ◽  
Orthonormal Basis ◽  
Trace Class ◽  
Linear Operators ◽  
Inner Product ◽  
Bounded Linear ◽  
Infinite Dimensional ◽  
Infinite Dimensional Hilbert Space ◽  
Schmidt Norm ◽  
Image Position

Let H denote a separable, infinite dimensional Hilbert space. Let B(H), C2 and C1 denote the algebra of all bounded linear operators acting on H, the Hilbert–Schmidt class and the trace class in B(H) respectively. It is well known that C2 and C1 each form a two-sided-ideal in B(H) and C2 is itself a Hilbert space with the inner productwhere {ei} is any orthonormal basis of H and tr(.) is the natural trace on C1. The Hilbert–Schmidt norm of X ∈ C2 is given by ⅡXⅡ2=(X, X)½.

Eigenfunction expansions on arbitrary domains

2005 ◽  
Vol 461 (2059) ◽  
pp. 2121-2133 ◽  
Author(s):  
P.N Shankar
Keyword(s):  
Classical Method ◽  
Boundary Value ◽  
Linear Partial Differential Equation ◽  
Three Dimensions ◽  
Inner Product ◽  
Eigenfunction Expansions ◽  
Complete Set ◽  
Coordinate Lines ◽  
Steady Heat Conduction ◽  
Steady Heat

Consider the boundary-value problem for the field ψ ( x ) which satisfies the linear partial differential equation in an arbitrary domain with data given on the boundary . It is generally believed that, unless is the union of constant coordinate lines in a separable coordinate system for the operator , the problem cannot be solved by the classical method of eigenfunction expansions. We show how this apparent limitation can be overcome. The key idea is to embed in a larger embedding domain , which is endowed with a complete set of eigenfunctions of the operator , where the λ n are the eigenvalues. We can now expand ψ ( x ) in terms of this set, i.e. . Although the unknown scalars { a n } can no longer be determined by the use of an inner product, a least-squares procedure which minimizes the error in the boundary data yields the scalars to as high a precision, in principle, as needed. Examples are given of steady heat conduction in two and three dimensions, governed by Laplace's equation, and of Stokes flow in a container, governed by the biharmonic equation, all in non-simple domains. The scope of a powerful classical method has, by this extension, been enlarged very considerably. It is believed that it will be of great use in solving practical, linear boundary-value problems, which until now had to be solved by brute force numerical methods.

scholarly journals Topological Quasilinear Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Yılmaz Yılmaz ◽  
Sümeyye Çakan ◽  
Şahika Aytekin
Keyword(s):  
Functional Analysis ◽  
Linear Functional ◽  
Vector Spaces ◽  
Multivalued Mappings ◽  
Linear Spaces ◽  
Localization Principle ◽  
Topological Vector Spaces

We introduce, in this work, the notion of topological quasilinear spaces as a generalization of the notion of normed quasilinear spaces defined by Aseev (1986). He introduced a kind of the concept of a quasilinear spaces both including a classical linear spaces and also nonlinear spaces of subsets and multivalued mappings. Further, Aseev presented some basic quasilinear counterpart of linear functional analysis by introducing the notions of norm and bounded quasilinear operators and functionals. Our investigations show that translation may destroy the property of being a neighborhood of a set in topological quasilinear spaces in contrast to the situation in topological vector spaces. Thus, we prove that any topological quasilinear space may not satisfy the localization principle of topological vector spaces.

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