|
Strong Operator Topology
In functional analysis, a branch of mathematics, the strong operator topology, often abbreviated SOT, is the locally convex topology on the set of bounded operators on a Hilbert space ''H'' induced by the seminorms of the form T\mapsto\, Tx\, , as ''x'' varies in ''H''. Equivalently, it is the coarsest topology such that, for each fixed ''x'' in ''H'', the evaluation map T\mapsto Tx (taking values in ''H'') is continuous in T. The equivalence of these two definitions can be seen by observing that a subbase for both topologies is given by the sets U(T_0,x,\epsilon) = \ (where ''T0'' is any bounded operator on ''H'', ''x'' is any vector and ε is any positive real number). In concrete terms, this means that T_i\to T in the strong operator topology if and only if \, T_ix-Tx\, \to 0 for each ''x'' in ''H''. The SOT is stronger than the weak operator topology and weaker than the norm topology. The SOT lacks some of the nicer properties that the weak operator topology has, but bein ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Functional Analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined on these spaces and respecting these structures in a suitable sense. The historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous, unitary etc. operators between function spaces. This point of view turned out to be particularly useful for the study of differential and integral equations. The usage of the word '' functional'' as a noun goes back to the calculus of variations, implying a function whose argument is a function. The term was first used in Hadamard's 1910 book on that subject. However, the general concept of a functional had previously been introduced in 1887 by the I ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weak Operator Topology
In functional analysis, the weak operator topology, often abbreviated WOT, is the weakest topology on the set of bounded operators on a Hilbert space H, such that the functional sending an operator T to the complex number \langle Tx, y\rangle is continuous for any vectors x and y in the Hilbert space. Explicitly, for an operator T there is base of neighborhoods of the following type: choose a finite number of vectors x_i, continuous functionals y_i, and positive real constants \varepsilon_i indexed by the same finite set I. An operator S lies in the neighborhood if and only if , y_i(T(x_i) - S(x_i)), 0. Relationships between different topologies on ''B(X,Y)'' The different terminology for the various topologies on B(X,Y) can sometimes be confusing. For instance, "strong convergence" for vectors in a normed space sometimes refers to norm-convergence, which is very often distinct from (and stronger than) than SOT-convergence when the normed space in question is B(X,Y). The wea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topologies On The Set Of Operators On A Hilbert Space
In the mathematical field of functional analysis there are several standard topologies which are given to the algebra of bounded linear operators on a Banach space . Introduction Let (T_n)_ be a sequence of linear operators on the Banach space . Consider the statement that (T_n)_ converges to some operator on . This could have several different meanings: * If \, T_n - T\, \to 0, that is, the operator norm of T_n - T (the supremum of \, T_n x - T x \, _X, where ranges over the unit ball in ) converges to 0, we say that T_n \to T in the uniform operator topology. * If T_n x \to Tx for all x \in X, then we say T_n \to T in the strong operator topology. * Finally, suppose that for all we have T_n x \to Tx in the weak topology of . This means that F(T_n x) \to F(T x) for all linear functionals on . In this case we say that T_n \to T in the weak operator topology. List of topologies on B(''H'') There are many topologies that can be defined on besides the ones used abov ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Strongly Continuous Semigroup
In mathematics, a ''C''0-semigroup, also known as a strongly continuous one-parameter semigroup, is a generalization of the exponential function. Just as exponential functions provide solutions of scalar linear constant coefficient ordinary differential equations, strongly continuous semigroups provide solutions of linear constant coefficient ordinary differential equations in Banach spaces. Such differential equations in Banach spaces arise from e.g. delay differential equations and partial differential equations. Formally, a strongly continuous semigroup is a representation of the semigroup (R+,+) on some Banach space ''X'' that is continuous in the strong operator topology. Thus, strictly speaking, a strongly continuous semigroup is not a semigroup, but rather a continuous representation of a very particular semigroup. Formal definition A strongly continuous semigroup on a Banach space X is a map T : \mathbb_+ \to L(X) such that # T(0) = I , (identity operator on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Set
In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex region is a subset that intersects every line into a single line segment (possibly empty). For example, a solid cube is a convex set, but anything that is hollow or has an indent, for example, a crescent shape, is not convex. The boundary of a convex set is always a convex curve. The intersection of all the convex sets that contain a given subset of Euclidean space is called the convex hull of . It is the smallest convex set containing . A convex function is a real-valued function defined on an interval with the property that its epigraph (the set of points on or above the graph of the function) is a convex set. Convex minimization is a subfield of optimization that studies the problem of minimizing convex functions over convex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Functional
In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers). If is a vector space over a field , the set of all linear functionals from to is itself a vector space over with addition and scalar multiplication defined pointwise. This space is called the dual space of , or sometimes the algebraic dual space, when a topological dual space is also considered. It is often denoted , p. 19, §3.1 or, when the field is understood, V^*; other notations are also used, such as V', V^ or V^. When vectors are represented by column vectors (as is common when a basis is fixed), then linear functionals are represented as row vectors, and their values on specific vectors are given by matrix products (with the row vector on the left). Examples * The constant zero function, mapping every vector to zero, is trivially a linear functional. * Index ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuous Functional Calculus
In mathematics, particularly in operator theory and C*-algebra theory, a continuous functional calculus is a functional calculus which allows the application of a continuous function to normal elements of a C*-algebra. Theorem Theorem. Let ''x'' be a normal element of a C*-algebra ''A'' with an identity element e. Let ''C'' be the C*-algebra of the bounded continuous functions on the spectrum σ(''x'') of ''x''. Then there exists a unique mapping π : C → A, where ''π(f)'' is denoted ''f(x)'', such that π is a unit-preserving morphism of C*-algebras and π(1) = e and π(id) = ''x'', where id denotes the function ''z'' → ''z'' on σ(''x''). In particular, this theorem implies that bounded normal operators on a Hilbert space have a continuous functional calculus. Its proof is almost immediate from the Gelfand representation: it suffices to assume ''A'' is the C*-algebra of continuous functions on some compact space ''X'' and define : \pi(f) = f \circ x. Uniqueness fol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Measurable Functional Calculus
In functional analysis, a branch of mathematics, the Borel functional calculus is a ''functional calculus'' (that is, an assignment of operators from commutative algebras to functions defined on their spectra), which has particularly broad scope. Thus for instance if ''T'' is an operator, applying the squaring function ''s'' → ''s''2 to ''T'' yields the operator ''T''2. Using the functional calculus for larger classes of functions, we can for example define rigorously the "square root" of the (negative) Laplacian operator or the exponential e^. The 'scope' here means the kind of ''function of an operator'' which is allowed. The Borel functional calculus is more general than the continuous functional calculus, and its focus is different than the holomorphic functional calculus one. More precisely, the Borel functional calculus allows for applying an arbitrary Borel function to a self-adjoint operator, in a way that generalizes applying a polynomial function. Motivation If ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Operator Norm
In mathematics, the operator norm measures the "size" of certain linear operators by assigning each a real number called its . Formally, it is a norm defined on the space of bounded linear operators between two given normed vector spaces. Introduction and definition Given two normed vector spaces V and W (over the same base field, either the real numbers \R or the complex numbers \Complex), a linear map A : V \to W is continuous if and only if there exists a real number c such that \, Av\, \leq c \, v\, \quad \mbox v\in V. The norm on the left is the one in W and the norm on the right is the one in V. Intuitively, the continuous operator A never increases the length of any vector by more than a factor of c. Thus the image of a bounded set under a continuous operator is also bounded. Because of this property, the continuous linear operators are also known as bounded operators. In order to "measure the size" of A, one can take the infimum of the numbers c such that the above ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finer Topology
In topology and related areas of mathematics, the set of all possible topologies on a given set forms a partially ordered set. This order relation can be used for comparison of the topologies. Definition A topology on a set may be defined as the collection of subsets which are considered to be "open". An alternative definition is that it is the collection of subsets which are considered "closed". These two ways of defining the topology are essentially equivalent because the complement of an open set is closed and vice versa. In the following, it doesn't matter which definition is used. Let ''τ''1 and ''τ''2 be two topologies on a set ''X'' such that ''τ''1 is contained in ''τ''2: :\tau_1 \subseteq \tau_2. That is, every element of ''τ''1 is also an element of ''τ''2. Then the topology ''τ''1 is said to be a coarser (weaker or smaller) topology than ''τ''2, and ''τ''2 is said to be a finer (stronger or larger) topology than ''τ''1. There are some authors, especiall ... [...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 t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subbase
In topology, a subbase (or subbasis, prebase, prebasis) for a topological space X with topology T is a subcollection B of T that generates T, in the sense that T is the smallest topology containing B. A slightly different definition is used by some authors, and there are other useful equivalent formulations of the definition; these are discussed below. Definition Let X be a topological space with topology T. A subbase of T is usually defined as a subcollection B of T satisfying one of the two following equivalent conditions: #The subcollection B ''generates'' the topology T. This means that T is the smallest topology containing B: any topology T^\prime on X containing B must also contain T. #The collection of open sets consisting of all finite intersections of elements of B, together with the set X, forms a basis for T. This means that every proper open set in T can be written as a union of finite intersections of elements of B. Explicitly, given a point x in an open set U \su ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
