![arXiv:math/0304461v1 [math.DS] 28 Apr 2003](http://s1.studyres.com/store/data/017912608_1-01ea405b40d94f16985213ba7f480945-300x300.png)
FINITE METRIC SPACES AND THEIR EMBEDDING INTO
... 2. The topology induced by d∗ on X/∼ forms K(X). For the topology induced by d∗ on X/∼ to be K(X), the equivalence classes must be comprised of topologically indistinguishable points. Take x,y ∈ X, with x and y topologically distinguishable. Then there is an open subset U of X where x ∈ U but y ∈ / ...
... 2. The topology induced by d∗ on X/∼ forms K(X). For the topology induced by d∗ on X/∼ to be K(X), the equivalence classes must be comprised of topologically indistinguishable points. Take x,y ∈ X, with x and y topologically distinguishable. Then there is an open subset U of X where x ∈ U but y ∈ / ...
Linear Transformations
... If we add these two equations we are left with 2x1 = 2x2 , so the first coordinates of the inputs are equal. Subracting the equations shows that the input’s second coordinates are also equal, and thus u = v. This constitutes a proof that the function given in problem 7 is one-to-one. The proof that ...
... If we add these two equations we are left with 2x1 = 2x2 , so the first coordinates of the inputs are equal. Subracting the equations shows that the input’s second coordinates are also equal, and thus u = v. This constitutes a proof that the function given in problem 7 is one-to-one. The proof that ...
Bochner vs. Pettis norm: examples and results
... Theorem to construct, for an arbitrary in nite-dimensional Banach space, a sequence of Bochner integrable functions whose Bochner norms tend to in nity but whose Pettis norms tend to zero. By re ning this example (again working with an arbitrary in nite-dimensional Banach space), we produce a Pettis ...
... Theorem to construct, for an arbitrary in nite-dimensional Banach space, a sequence of Bochner integrable functions whose Bochner norms tend to in nity but whose Pettis norms tend to zero. By re ning this example (again working with an arbitrary in nite-dimensional Banach space), we produce a Pettis ...
Integral Vector Theorems - Queen`s University Belfast
... (a) dS is a unit normal pointing outwards from the interior of the volume V . (b) Both sides of the equation are scalars. (c) The theorem is often a useful way of calculating a surface integral over a surface composed of several distinct parts (e.g. a cube). (d) ∇ · F is a scalar field representing ...
... (a) dS is a unit normal pointing outwards from the interior of the volume V . (b) Both sides of the equation are scalars. (c) The theorem is often a useful way of calculating a surface integral over a surface composed of several distinct parts (e.g. a cube). (d) ∇ · F is a scalar field representing ...
![arXiv:math/0005256v2 [math.QA] 21 Jun 2000](http://s1.studyres.com/store/data/017029559_1-d28b6a03fef36ff14eaa2fb575ab89be-300x300.png)
Introduction to representation theory of finite groups
... 5. Let G be a finite group and let X = {x1 , . . . , xn } be a finite set on which G acts. Over any field K, consider an n-dimensional vector space V with a basis indexed by elements of X: vx1 , . . . , vxn . We can let G act on V by g(vx ) = vg(x) for all x ∈ X and g ∈ G. Thus, G permutes the basi ...
... 5. Let G be a finite group and let X = {x1 , . . . , xn } be a finite set on which G acts. Over any field K, consider an n-dimensional vector space V with a basis indexed by elements of X: vx1 , . . . , vxn . We can let G act on V by g(vx ) = vg(x) for all x ∈ X and g ∈ G. Thus, G permutes the basi ...
Orientation and degree for Fredholm maps of index zero between
... Let us now define a notion of orientation for nonlinear Fredholm maps of index zero between open subsets of Banach spaces. Definition 2.6 Let Ω be an open subset of E and f : Ω → F be Fredholm of index zero. An orientation of f is an orientation of Df : x 7→ Df (x) ∈ Φ0 (E, F ), and f is orientable ...
... Let us now define a notion of orientation for nonlinear Fredholm maps of index zero between open subsets of Banach spaces. Definition 2.6 Let Ω be an open subset of E and f : Ω → F be Fredholm of index zero. An orientation of f is an orientation of Df : x 7→ Df (x) ∈ Φ0 (E, F ), and f is orientable ...
Vector Bundles and K-Theory
... Topological K–theory, the first generalized cohomology theory to be studied thoroughly, was introduced around 1960 by Atiyah and Hirzebruch, based on the Periodicity Theorem of Bott proved just a few years earlier. In some respects K–theory is more elementary than classical homology and cohomology, ...
... Topological K–theory, the first generalized cohomology theory to be studied thoroughly, was introduced around 1960 by Atiyah and Hirzebruch, based on the Periodicity Theorem of Bott proved just a few years earlier. In some respects K–theory is more elementary than classical homology and cohomology, ...
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.