A Special Partial order on Interval Normed Spaces
... problems. At first, the concepts of interval spaces and nonstandard normed interval spaces are introducted. In Interval normed spaces are spaces that have less known in comparison with other spaces. Recently, Many authors have been interested in such spaces; For example, in [1] have defined interval ...
... problems. At first, the concepts of interval spaces and nonstandard normed interval spaces are introducted. In Interval normed spaces are spaces that have less known in comparison with other spaces. Recently, Many authors have been interested in such spaces; For example, in [1] have defined interval ...
Angles and a Classification of Normed Spaces
... In this paper we deal with generalized real normed vector spaces. We consider vector spaces X provided with a ‘weight’ or ‘functional’ k · k, that means we have a continuous map k · k : X −→ + ∪ {0}. We assume that the weights are ‘absolute homogeneous’ or ‘balanced’, i.e. kr · ~xk = |r| · k~xk for ...
... In this paper we deal with generalized real normed vector spaces. We consider vector spaces X provided with a ‘weight’ or ‘functional’ k · k, that means we have a continuous map k · k : X −→ + ∪ {0}. We assume that the weights are ‘absolute homogeneous’ or ‘balanced’, i.e. kr · ~xk = |r| · k~xk for ...
5 Unitary groups
... points of this type (which are obviously determined by any two of their members) will be the H-lines. It is readily checked that the H-lines do indeed arise in the manner described in Section 4.4, that is, as the sets of points of Ω orthogonal to two given non-orthogonal points. So condition (b) hol ...
... points of this type (which are obviously determined by any two of their members) will be the H-lines. It is readily checked that the H-lines do indeed arise in the manner described in Section 4.4, that is, as the sets of points of Ω orthogonal to two given non-orthogonal points. So condition (b) hol ...
Math for Machine Learning
... 4x3 + 3x2 − 4x and the second derivative is 12x2 + 6x − 4. It’s fairly easy to find a value of x for which the second derivative is negative: 0 is such an example. It is moderately interesting to note that while this f is not convex everywhere, it is convex in certain ranges, for instance the open i ...
... 4x3 + 3x2 − 4x and the second derivative is 12x2 + 6x − 4. It’s fairly easy to find a value of x for which the second derivative is negative: 0 is such an example. It is moderately interesting to note that while this f is not convex everywhere, it is convex in certain ranges, for instance the open i ...
1/16/15
... Proof. We will prove that for all n less than or equal to the cardinality of I (which may be all n, if I is infinite) that any subset {vi1 , . . . , vin } of size n of {vi }i∈I is linearly independent. This is clearly true for n = 1, since eigenvectors are nonzero by definition. Now suppose that it’ ...
... Proof. We will prove that for all n less than or equal to the cardinality of I (which may be all n, if I is infinite) that any subset {vi1 , . . . , vin } of size n of {vi }i∈I is linearly independent. This is clearly true for n = 1, since eigenvectors are nonzero by definition. Now suppose that it’ ...
1 VECTOR SPACES AND SUBSPACES
... • A set W of one or more vectors from a vector space V is said to be closed under addition if condition (a) in theorem 1.4 holds and closed under scalar multiplication if condition (b) holds. Thus, theorem 1.4 states that W is a subspace of V if and only if W is closed under addition and closed unde ...
... • A set W of one or more vectors from a vector space V is said to be closed under addition if condition (a) in theorem 1.4 holds and closed under scalar multiplication if condition (b) holds. Thus, theorem 1.4 states that W is a subspace of V if and only if W is closed under addition and closed unde ...
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.