Chapter 3: Weak topologies and Tychonoff`s theorem
... E ∗∗ = (E ∗ )∗ . That is because the weak-topology on E is given by the seminorms pα : E → [0, ∞) with pα (x) = |α(x)| with α ∈ E ∗ and the weak*-topology on (E ∗ )∗ is also given by seminorms given by α ∈ E ∗ . To make the notation clear we might denote the seminorms on E ∗∗ by p̃α and write p̃α (z ...
... E ∗∗ = (E ∗ )∗ . That is because the weak-topology on E is given by the seminorms pα : E → [0, ∞) with pα (x) = |α(x)| with α ∈ E ∗ and the weak*-topology on (E ∗ )∗ is also given by seminorms given by α ∈ E ∗ . To make the notation clear we might denote the seminorms on E ∗∗ by p̃α and write p̃α (z ...
Chapter 3 Linear Codes
... code C in Fqn is a q n−k × q k array listing all the cosets of Fqn in which the first row consists of the code C with 0 on the extreme left, and the other rows are the cosets ui + C, each arranged in corresponding order, with the coset leader on the left. Remark. The standard array may be constructe ...
... code C in Fqn is a q n−k × q k array listing all the cosets of Fqn in which the first row consists of the code C with 0 on the extreme left, and the other rows are the cosets ui + C, each arranged in corresponding order, with the coset leader on the left. Remark. The standard array may be constructe ...
CONTINUOUS SELECTIONS AND FIXED POINTS OF MULTI
... Let X and Y be topological spaces. A multi-valued map T : X → 2Y is a map from X into the power set 2Y of Y . Let T −1 : Y → 2X be defined by the condition that x ∈ T −1 y if and only if y ∈ T (x). Recall that (a) T is said to be closed if its graph Gr (T ) = {(x, y) : x ∈ X, y ∈ T (x)} is closed in ...
... Let X and Y be topological spaces. A multi-valued map T : X → 2Y is a map from X into the power set 2Y of Y . Let T −1 : Y → 2X be defined by the condition that x ∈ T −1 y if and only if y ∈ T (x). Recall that (a) T is said to be closed if its graph Gr (T ) = {(x, y) : x ∈ X, y ∈ T (x)} is closed in ...
The ideal center of partially ordered vector spaces
... then [ E K . This shows that the evaluation m a p A: a EZ~-+Aa E C(K) with Aa(/)=/(a), / E K is a bipositive, isometric map. An element 0 =~/E K is extremal iff II/1[= 1 and 0 ~
... then [ E K . This shows that the evaluation m a p A: a EZ~-+Aa E C(K) with Aa(/)=/(a), / E K is a bipositive, isometric map. An element 0 =~/E K is extremal iff II/1[= 1 and 0 ~
AN AXIOMATIC FORMULATION OF QUANTUM MECHANICS HONORS THESIS ITHACA COLLEGE DEPARTMENT OF MATHEMATICS
... When David Hilbert called for the axiomatization of physical theories as part of his notable twenty-three problems, he singled out in particular the applications of probability theory to statistical mechanics and of Lie theory to classical mechanics. However, underpinning these specific examples was ...
... When David Hilbert called for the axiomatization of physical theories as part of his notable twenty-three problems, he singled out in particular the applications of probability theory to statistical mechanics and of Lie theory to classical mechanics. However, underpinning these specific examples was ...
FINITE SEMIFIELDS WITH A LARGE NUCLEUS AND HIGHER
... semifields is equivalent with an (n − 1)-dimensional subspace skew to a determinantal hypersurface in PG(n2 − 1, q), and provided an answer to the isotopism problem in [2]. In this paper we give a generalisation of the BEL-construction using linear sets, and then concentrate on this configuration an ...
... semifields is equivalent with an (n − 1)-dimensional subspace skew to a determinantal hypersurface in PG(n2 − 1, q), and provided an answer to the isotopism problem in [2]. In this paper we give a generalisation of the BEL-construction using linear sets, and then concentrate on this configuration an ...
Notes on Classical Groups - School of Mathematical Sciences
... Exercise 1.11 Prove the above assertions. It is usual to be less formal with the language of incidence, and say “the point P lies on the line L”, or “the line L passes through the point P” rather than “the point P and the line L are incident”. Similar geometric language will be used without further ...
... Exercise 1.11 Prove the above assertions. It is usual to be less formal with the language of incidence, and say “the point P lies on the line L”, or “the line L passes through the point P” rather than “the point P and the line L are incident”. Similar geometric language will be used without further ...
Vector space
A vector space (also called a linear space) is a collection of objects called vectors, which may be added together and multiplied (""scaled"") by numbers, called scalars in this context. Scalars are often taken to be real numbers, but there are also vector spaces with scalar multiplication by complex numbers, rational numbers, or generally any field. The operations of vector addition and scalar multiplication must satisfy certain requirements, called axioms, listed below. Euclidean vectors are an example of a vector space. They represent physical quantities such as forces: any two forces (of the same type) can be added to yield a third, and the multiplication of a force vector by a real multiplier is another force vector. In the same vein, but in a more geometric sense, vectors representing displacements in the plane or in three-dimensional space also form vector spaces. Vectors in vector spaces do not necessarily have to be arrow-like objects as they appear in the mentioned examples: vectors are regarded as abstract mathematical objects with particular properties, which in some cases can be visualized as arrows.Vector spaces are the subject of linear algebra and are well understood from this point of view since vector spaces are characterized by their dimension, which, roughly speaking, specifies the number of independent directions in the space. A vector space may be endowed with additional structure, such as a norm or inner product. Such spaces arise naturally in mathematical analysis, mainly in the guise of infinite-dimensional function spaces whose vectors are functions. Analytical problems call for the ability to decide whether a sequence of vectors converges to a given vector. This is accomplished by considering vector spaces with additional structure, mostly spaces endowed with a suitable topology, thus allowing the consideration of proximity and continuity issues. These topological vector spaces, in particular Banach spaces and Hilbert spaces, have a richer theory.Historically, the first ideas leading to vector spaces can be traced back as far as the 17th century's analytic geometry, matrices, systems of linear equations, and Euclidean vectors. The modern, more abstract treatment, first formulated by Giuseppe Peano in 1888, encompasses more general objects than Euclidean space, but much of the theory can be seen as an extension of classical geometric ideas like lines, planes and their higher-dimensional analogs.Today, vector spaces are applied throughout mathematics, science and engineering. They are the appropriate linear-algebraic notion to deal with systems of linear equations; offer a framework for Fourier expansion, which is employed in image compression routines; or provide an environment that can be used for solution techniques for partial differential equations. Furthermore, vector spaces furnish an abstract, coordinate-free way of dealing with geometrical and physical objects such as tensors. This in turn allows the examination of local properties of manifolds by linearization techniques. Vector spaces may be generalized in several ways, leading to more advanced notions in geometry and abstract algebra.