Subspace mixing and universality criterion for a sequence of operators
... Let B(X) denote the algebra of all bounded linear operators on a infinite-dimensional separable complex Banach space X. For x ∈ X, the orbit of x under T is the set Orb(T, x) = {T n x : n ∈ N}. A vector x is called hypercyclic for T if Orb(T, x) is dense in X and the operator T is said to be hypercy ...
... Let B(X) denote the algebra of all bounded linear operators on a infinite-dimensional separable complex Banach space X. For x ∈ X, the orbit of x under T is the set Orb(T, x) = {T n x : n ∈ N}. A vector x is called hypercyclic for T if Orb(T, x) is dense in X and the operator T is said to be hypercy ...
Number of subgroups of index 2.
... In fact, it is easily seen that the set on the right is non-empty and closed under multiplication. The closure under inverses follows from (x1 · · · xn ) 1 = xn 1 · · · x1 1 , and the fact that since G is finite each xi 1 is equal to xki i for some ki 1. Thus, by the two-step subgroup test ([3, Theo ...
... In fact, it is easily seen that the set on the right is non-empty and closed under multiplication. The closure under inverses follows from (x1 · · · xn ) 1 = xn 1 · · · x1 1 , and the fact that since G is finite each xi 1 is equal to xki i for some ki 1. Thus, by the two-step subgroup test ([3, Theo ...
here
... The boundary maps πn (B, b) → πn−1 (F, x) are defined by using the fibration property to lift a map of pairs (Dn , S n−1 ) → (B, b) to (E, F ) extending the lift of basepoints s → b lifted to s → x. Exactness at each term is proved easily by the homotopy lifting property for disks and spheres. From ...
... The boundary maps πn (B, b) → πn−1 (F, x) are defined by using the fibration property to lift a map of pairs (Dn , S n−1 ) → (B, b) to (E, F ) extending the lift of basepoints s → b lifted to s → x. Exactness at each term is proved easily by the homotopy lifting property for disks and spheres. From ...
Linear Transformations and Matrices
... and the kernel of T to be the set Ker T = {x ∞ U: T(x) = 0} . (Many authors call Im T the range of T, but we use this term to mean the space V in which T takes its values.) Since T(0) = 0 ∞ V, we see that 0 ∞ Im T, and hence Im T ≠ Å. Now suppose xæ, yæ ∞ Im T. Then there exist x, y ∞ U such that T( ...
... and the kernel of T to be the set Ker T = {x ∞ U: T(x) = 0} . (Many authors call Im T the range of T, but we use this term to mean the space V in which T takes its values.) Since T(0) = 0 ∞ V, we see that 0 ∞ Im T, and hence Im T ≠ Å. Now suppose xæ, yæ ∞ Im T. Then there exist x, y ∞ U such that T( ...
Math 121A Linear Algebra
... Definition: A subset S V, V a vector space, is called linearly dependent if and only if a finite number of vectors u1, …, un S and a finite number of scalars a1, …, an F (not all zero) such that a1u1 + … + anun = 0, a non-trivial representation of the zero vector. Remark: Trivially, if ai= 0 ...
... Definition: A subset S V, V a vector space, is called linearly dependent if and only if a finite number of vectors u1, …, un S and a finite number of scalars a1, …, an F (not all zero) such that a1u1 + … + anun = 0, a non-trivial representation of the zero vector. Remark: Trivially, if ai= 0 ...
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.