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Topological Divisor Of Zero
In mathematics, an element z of a Banach algebra A is called a topological divisor of zero if there exists a sequence x_1,x_2,x_3,... of elements of A such that # The sequence zx_n converges to the zero element, but # The sequence x_n does not converge to the zero element. If such a sequence exists, then one may assume that \left \Vert \ x_n \right \, = 1 for all n. If A is not commutative, then z is called a "left" topological divisor of zero, and one may define "right" topological divisors of zero similarly. Examples * If A has a unit element, then the invertible elements of A form an open subset of A, while the non-invertible elements are the complementary closed subset. Any point on the boundary between these two sets is both a left and right topological divisor of zero. * In particular, any quasinilpotent element is a topological divisor of zero (e.g. the Volterra operator). * An operator on a Banach space X, which is injective, not surjective, but whose image is dense in X, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Banach Algebra
In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers (or over a non-Archimedean complete normed field) that at the same time is also a Banach space, that is, a normed space that is complete in the metric induced by the norm. The norm is required to satisfy \, x \, y\, \ \leq \, x\, \, \, y\, \quad \text x, y \in A. This ensures that the multiplication operation is continuous. A Banach algebra is called ''unital'' if it has an identity element for the multiplication whose norm is 1, and ''commutative'' if its multiplication is commutative. Any Banach algebra A (whether it has an identity element or not) can be embedded isometrically into a unital Banach algebra A_e so as to form a closed ideal of A_e. Often one assumes ''a priori'' that the algebra under consideration is unital: for one can develop much of the theory by considering A_e and then applying the outcome in the ori ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called the ''length'' of the sequence. Unlike a set, the same elements can appear multiple times at different positions in a sequence, and unlike a set, the order does matter. Formally, a sequence can be defined as a function from natural numbers (the positions of elements in the sequence) to the elements at each position. The notion of a sequence can be generalized to an indexed family, defined as a function from an ''arbitrary'' index set. For example, (M, A, R, Y) is a sequence of letters with the letter 'M' first and 'Y' last. This sequence differs from (A, R, M, Y). Also, the sequence (1, 1, 2, 3, 5, 8), which contains the number 1 at two different positions, is a valid sequence. Sequences can be ''finite'', as in these examples, or ''infi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Commutativity
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of the property that says something like or , the property can also be used in more advanced settings. The name is needed because there are operations, such as division and subtraction, that do not have it (for example, ); such operations are ''not'' commutative, and so are referred to as ''noncommutative operations''. The idea that simple operations, such as the multiplication and addition of numbers, are commutative was for many years implicitly assumed. Thus, this property was not named until the 19th century, when mathematics started to become formalized. A similar property exists for binary relations; a binary relation is said to be symmetric if the relation applies regardless of the order of its operands; for example, equality is s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Open Set
In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are sufficiently near to (that is, all points whose distance to is less than some value depending on ). More generally, one defines open sets as the members of a given collection of subsets of a given set, a collection that has the property of containing every union of its members, every finite intersection of its members, the empty set, and the whole set itself. A set in which such a collection is given is called a topological space, and the collection is called a topology. These conditions are very loose, and allow enormous flexibility in the choice of open sets. For example, ''every'' subset can be open (the discrete topology), or no set can be open except the space itself and the empty set (the indiscrete topology). In practice, however, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Closed Set
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a closed set is a set which is closed under the limit operation. This should not be confused with a closed manifold. Equivalent definitions By definition, a subset A of a topological space (X, \tau) is called if its complement X \setminus A is an open subset of (X, \tau); that is, if X \setminus A \in \tau. A set is closed in X if and only if it is equal to its closure in X. Equivalently, a set is closed if and only if it contains all of its limit points. Yet another equivalent definition is that a set is closed if and only if it contains all of its boundary points. Every subset A \subseteq X is always contained in its (topological) closure in X, which is denoted by \operatorname_X A; that is, if A \subseteq X then A \subseteq \oper ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boundary (topology)
In topology and mathematics in general, the boundary of a subset of a topological space is the set of points in the closure of not belonging to the interior of . An element of the boundary of is called a boundary point of . The term boundary operation refers to finding or taking the boundary of a set. Notations used for boundary of a set include \operatorname(S), \operatorname(S), and \partial S. Some authors (for example Willard, in ''General Topology'') use the term frontier instead of boundary in an attempt to avoid confusion with a different definition used in algebraic topology and the theory of manifolds. Despite widespread acceptance of the meaning of the terms boundary and frontier, they have sometimes been used to refer to other sets. For example, ''Metric Spaces'' by E. T. Copson uses the term boundary to refer to Hausdorff's border, which is defined as the intersection of a set with its boundary. Hausdorff also introduced the term residue, which is defi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quasinilpotent
In operator theory, a bounded operator ''T'' on a Hilbert space is said to be nilpotent if ''Tn'' = 0 for some ''n''. It is said to be quasinilpotent or topologically nilpotent if its spectrum ''σ''(''T'') = . Examples In the finite-dimensional case, i.e. when ''T'' is a square matrix with complex entries, ''σ''(''T'') = if and only if ''T'' is similar to a matrix whose only nonzero entries are on the superdiagonal, by the Jordan canonical form. In turn this is equivalent to ''Tn'' = 0 for some ''n''. Therefore, for matrices, quasinilpotency coincides with nilpotency. This is not true when ''H'' is infinite-dimensional. Consider the Volterra operator, defined as follows: consider the unit square ''X'' = ,1× ,1⊂ R2, with the Lebesgue measure ''m''. On ''X'', define the (kernel) function ''K'' by :K(x,y) = \left\{ \begin{matrix} 1, & \mbox{if} \; x \geq y\\ 0, & \mbox{otherwise}. \end{matrix} \right. The Volterra operator is the corresponding integral ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Volterra Operator
In mathematics, in the area of functional analysis and operator theory, the Volterra operator, named after Vito Volterra, is a bounded linear operator on the space ''L''2 ,1of complex-valued square-integrable functions on the interval ,1 On the subspace ''C'' ,1of continuous functions it represents indefinite integration. It is the operator corresponding to the Volterra integral equations. Definition The Volterra operator, ''V'', may be defined for a function ''f'' ∈ ''L''2 ,1and a value ''t'' ∈ ,1 as :V(f)(t) = \int_^ f(s)\, ds. Properties *''V'' is a bounded linear operator between Hilbert spaces, with Hermitian adjoint V^*(f)(t) = \int_^ f(s)\, ds. *''V'' is a Hilbert–Schmidt operator, hence in particular is compact. *''V'' has no eigenvalues and therefore, by the spectral theory of compact operators, its spectrum ''σ''(''V'') = . *''V'' is a quasinilpotent operator (that is, the spectral radius, ''ρ''(''V''), is zero), but it is not nilpotent. *Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Injective
In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositive statement.) In other words, every element of the function's codomain is the image of one element of its domain. The term must not be confused with that refers to bijective functions, which are functions such that each element in the codomain is an image of exactly one element in the domain. A homomorphism between algebraic structures is a function that is compatible with the operations of the structures. For all common algebraic structures, and, in particular for vector spaces, an is also called a . However, in the more general context of category theory, the definition of a monomorphism differs from that of an injective homomorphism. This is thus a theorem that they are equivalent for algebraic structures; see for more details. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Surjective
In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of its domain. It is not required that be unique; the function may map one or more elements of to the same element of . The term ''surjective'' and the related terms ''injective'' and ''bijective'' were introduced by Nicolas Bourbaki, a group of mainly French 20th-century mathematicians who, under this pseudonym, wrote a series of books presenting an exposition of modern advanced mathematics, beginning in 1935. The French word '' sur'' means ''over'' or ''above'', and relates to the fact that the image of the domain of a surjective function completely covers the function's codomain. Any function induces a surjection by restricting its codomain to the image of its domain. Every surjective function has a right inverse assuming the axiom ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological Algebra
In mathematics, a topological algebra A is an algebra and at the same time a topological space, where the algebraic and the topological structures are coherent in a specified sense. Definition A topological algebra A over a topological field K is a topological vector space together with a bilinear multiplication :\cdot: A \times A \to A, :(a,b) \mapsto a \cdot b that turns A into an algebra over K and is continuous in some definite sense. Usually the ''continuity of the multiplication'' is expressed by one of the following (non-equivalent) requirements: * ''joint continuity'': for each neighbourhood of zero U\subseteq A there are neighbourhoods of zero V\subseteq A and W\subseteq A such that V \cdot W\subseteq U (in other words, this condition means that the multiplication is continuous as a map between topological spaces or * ''stereotype continuity'': for each totally bounded set S\subseteq A and for each neighbourhood of zero U\subseteq A there is a neighbourhood of zero V\su ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
