Functionals of Bounded Frechet Variation

1949 ◽  
Vol 1 (2) ◽  
pp. 153-165 ◽  
Author(s):  
Marston Morse ◽  
William Transue
Keyword(s):  
Representation Theory ◽  
Spectral Theory ◽  
General Type ◽  
Integrodifferential Equations ◽  
Vector Spaces ◽  
Variational Theory ◽  
Characteristic Equations ◽  
Special Cases ◽  
Jacobi Equations ◽  
Normed Vector

In a series of papers which will follow this paper the authors will present a theory of functionals which are bilinear over a product A × B of two normed vector spaces A and B. This theory will include a representation theory, a variational theory, and a spectral theory. The associated characteristic equations will include as special cases the Jacobi equations of the classical variational theory when n = 1, and self-adjoint integrodifferential equations of very general type. The bilinear theory is oriented by the needs of non-linear and non-bilinear analysis in the large.

scholarly journals Subdifferential Properties of Minimal Time Functions Associated with Set-Valued Mappings with Closed Convex Graphs in Hausdorff Topological Vector Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Messaoud Bounkhel
Keyword(s):  
Optimization Problems ◽  
Time Function ◽  
Vector Spaces ◽  
Directional Derivatives ◽  
Minimal Time Function ◽  
Topological Vector Spaces ◽  
Minimal Time ◽  
Set Valued Mapping ◽  
Bounded Set ◽  
Normed Vector

For a set-valued mappingMdefined between two Hausdorff topological vector spacesEandFand with closed convex graph and for a given point(x,y)∈E×F, we study the minimal time function associated with the images ofMand a bounded setΩ⊂Fdefined by𝒯M,Ω(x,y):=inf{t≥0:M(x)∩(y+tΩ)≠∅}. We prove and extend various properties on directional derivatives and subdifferentials of𝒯M,Ωat those points of(x,y)∈E×F(both cases: points in the graphgph Mand points outside the graph). These results are used to prove, in terms of the minimal time function, various new characterizations of the convex tangent cone and the convex normal cone to the graph ofMat points insidegph Mand to the graph of the enlargement set-valued mapping at points outsidegph M. Our results extend many existing results, from Banach spaces and normed vector spaces to Hausdorff topological vector spaces (Bounkhel, 2012; Bounkhel and Thibault, 2002; Burke et al., 1992; He and Ng, 2006; and Jiang and He 2009). An application of the minimal time function𝒯M,Ωto the calmness property of perturbed optimization problems in Hausdorff topological vector spaces is given in the last section of the paper.

A Review On Some Neutrosophic Algebraic Linear Structures

2021 ◽  
Author(s):  
Malath F. Alaswad ◽  
Keyword(s):  
Linear Algebra ◽  
Matrix Theory ◽  
Vector Spaces ◽  
Interested Reader ◽  
Linear Structures ◽  
Basic Concepts

This paper is dedicated to reviewing some of the basic concepts in neutrosophic linear algebra and its generalizations, especially neutrosophic vector spaces, refined neutrosophic, and n-refined neutrosophic vector spaces. Also, this work gives the interested reader a strong background in the study of neutrosophic matrix theory and n-refined neutrosophic matrix theory. We study elementary properties of these concepts such as Kernel, AH-Quotient, and dimension.

Linear Algebra and the Dirac Notation

2006 ◽  
Author(s):  
Phillip Kaye ◽  
Raymond Laflamme ◽  
Michele Mosca
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Quantum Computing ◽  
Linear Algebra ◽  
Hilbert Spaces ◽  
Vector Spaces ◽  
Complex Vector ◽  
Dual Vector ◽  
Finite Dimensional ◽  
Basic Facts

We assume the reader has a strong background in elementary linear algebra. In this section we familiarize the reader with the algebraic notation used in quantum mechanics, remind the reader of some basic facts about complex vector spaces, and introduce some notions that might not have been covered in an elementary linear algebra course. The linear algebra notation used in quantum computing will likely be familiar to the student of physics, but may be alien to a student of mathematics or computer science. It is the Dirac notation, which was invented by Paul Dirac and which is used often in quantum mechanics. In mathematics and physics textbooks, vectors are often distinguished from scalars by writing an arrow over the identifying symbol: e.g a⃗. Sometimes boldface is used for this purpose: e.g. a. In the Dirac notation, the symbol identifying a vector is written inside a ‘ket’, and looks like |a⟩. We denote the dual vector for a (defined later) with a ‘bra’, written as ⟨a|. Then inner products will be written as ‘bra-kets’ (e.g. ⟨a|b⟩). We now carefully review the definitions of the main algebraic objects of interest, using the Dirac notation. The vector spaces we consider will be over the complex numbers, and are finite-dimensional, which significantly simplifies the mathematics we need. Such vector spaces are members of a class of vector spaces called Hilbert spaces. Nothing substantial is gained at this point by defining rigorously what a Hilbert space is, but virtually all the quantum computing literature refers to a finite-dimensional complex vector space by the name ‘Hilbert space’, and so we will follow this convention. We will use H to denote such a space. Since H is finite-dimensional, we can choose a basis and alternatively represent vectors (kets) in this basis as finite column vectors, and represent operators with finite matrices. As you see in Section 3, the Hilbert spaces of interest for quantum computing will typically have dimension 2n, for some positive integer n. This is because, as with classical information, we will construct larger state spaces by concatenating a string of smaller systems, usually of size two.

Functionals F Bilinear over the Product A × B of Two P-Normed Vector Spaces

1949 ◽  
Vol 50 (4) ◽  
pp. 777 ◽  
Author(s):  
Marston Morse ◽  
William Transue
Keyword(s):  
Vector Spaces ◽  
Normed Vector

Uniform Domains and Uniform Domain Decomposition Property in Real Normed Vector Spaces

2013 ◽  
Vol 113 (1) ◽  
pp. 128 ◽  
Author(s):  
M. Huang ◽  
X. Wang
Keyword(s):  
Vector Space ◽  
Domain Decomposition ◽  
Vector Spaces ◽  
Uniform Domain ◽  
Normed Vector Space ◽  
Decomposition Property ◽  
Uniform Domains ◽  
Normed Vector

Let $E$ be a real normed vector space with $\dim(E)\geq 2$, $D$ a proper subdomain of $E$. In this paper we characterize uniform domains in $E$ in terms of the uniform domain decomposition property. In addition, we discuss the relation between quasiballs and domains with the quasiball decomposition property in $\mathsf{R}^n$.

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