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Closed Range Theorem
In the mathematical theory of Banach spaces, the closed range theorem gives necessary and sufficient conditions for a closed densely defined operator to have closed range. The theorem was proved by Stefan Banach in his 1932 '' Théorie des opérations linéaires''. Statement Let X and Y be Banach spaces, T : D(T) \to Y a closed linear operator whose domain D(T) is dense in X, and T' the transpose of T. The theorem asserts that the following conditions are equivalent: * R(T), the range of T, is closed in Y. * R(T'), the range of T', is closed in X', the dual of X. * R(T) = N(T')^\perp = \left\. * R(T') = N(T)^\perp = \left\. Where N(T) and N(T') are the null space of T and T', respectively. Note that there is always an inclusion R(T)\subseteq N(T')^\perp, because if y=Tx and x^*\in N(T'), then \langle x^*,y\rangle = \langle T'x^*,x\rangle = 0. Likewise, there is an inclusion R(T')\subseteq N(T)^\perp. So the non-trivial part of the above theorem is the opposite inclusion ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Banach Space
In mathematics, more specifically in functional analysis, a Banach space (, ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well-defined limit that is within the space. Banach spaces are named after the Polish mathematician Stefan Banach, who introduced this concept and studied it systematically in 1920–1922 along with Hans Hahn and Eduard Helly. Maurice René Fréchet was the first to use the term "Banach space" and Banach in turn then coined the term " Fréchet space". Banach spaces originally grew out of the study of function spaces by Hilbert, Fréchet, and Riesz earlier in the century. Banach spaces play a central role in functional analysis. In other areas of analysis, the spaces under study are often Banach spaces. Definition A Banach space is a complete nor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Closed Linear Operator
In functional analysis, a branch of mathematics, a closed linear operator or often a closed operator is a linear operator whose graph is closed (see closed graph property). It is a basic example of an unbounded operator. The closed graph theorem says a linear operator f : X \to Y between Banach spaces is a closed operator if and only if it is a bounded operator and the domain of the operator is X. Hence, a closed linear operator that is used in practice is typically only defined on a dense subspace of a Banach space. Definition It is common in functional analysis to consider partial functions, which are functions defined on a subset of some space X. A partial function f is declared with the notation f : D \subseteq X \to Y, which indicates that f has prototype f : D \to Y (that is, its domain is D and its codomain is Y) Every partial function is, in particular, a function and so all terminology for functions can be applied to them. For instance, the graph of a partial functi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Densely Defined Operator
In mathematics – specifically, in operator theory – a densely defined operator or partially defined operator is a type of partially defined function. In a topological sense, it is a linear operator that is defined "almost everywhere". Densely defined operators often arise in functional analysis as operations that one would like to apply to a larger class of objects than those for which they ''a priori'' "make sense". A closed operator that is used in practice is often densely defined. Definition A densely defined linear operator T from one topological vector space, X, to another one, Y, is a linear operator that is defined on a dense linear subspace \operatorname(T) of X and takes values in Y, written T : \operatorname(T) \subseteq X \to Y. Sometimes this is abbreviated as T : X \to Y when the context makes it clear that X might not be the set-theoretic domain of T. Examples Consider the space C^0(, 1 \R) of all real-valued, continuous functions defined on the unit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Closed Set
In geometry, topology, and related branches of mathematics, a closed set is a Set (mathematics), set whose complement (set theory), 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 Closure (mathematics), closed under the limit of a sequence, limit operation. This should not be confused with closed manifold. Sets that are both open and closed and are called clopen sets. Definition Given a topological space (X, \tau), the following statements are equivalent: # a set A \subseteq X is in X. # A^c = X \setminus A is an open subset of (X, \tau); that is, A^ \in \tau. # A is equal to its Closure (topology), closure in X. # A contains all of its limit points. # A contains all of its Boundary (topology), boundary points. An alternative characterization (mathematics), characterization of closed sets is available via sequences and Net (mathematics), net ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Range Of A Function
In mathematics, the range of a function may refer either to the codomain of the function, or the image of the function. In some cases the codomain and the image of a function are the same set; such a function is called ''surjective'' or ''onto''. For any non-surjective function f: X \to Y, the codomain Y and the image \tilde Y are different; however, a new function can be defined with the original function's image as its codomain, \tilde: X \to \tilde where \tilde(x) = f(x). This new function is surjective. Definitions Given two sets and , a binary relation between and is a function (from to ) if for every element in there is exactly one in such that relates to . The sets and are called the '' domain'' and ''codomain'' of , respectively. The ''image'' of the function is the subset of consisting of only those elements of such that there is at least one in with . Usage As the term "range" can have different meanings, it is considered a good practice to define ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stefan Banach
Stefan Banach ( ; 30 March 1892 – 31 August 1945) was a Polish mathematician who is generally considered one of the 20th century's most important and influential mathematicians. He was the founder of modern functional analysis, and an original member of the Lwów School of Mathematics. His major work was the 1932 book, ''Théorie des opérations linéaires'' (Theory of Linear Operations), the first monograph on the general theory of functional analysis. Born in Kraków to a family of Gorals, Goral descent, Banach showed a keen interest in mathematics and engaged in solving mathematical problems during school Recess (break), recess. After completing his secondary education, he befriended Hugo Steinhaus, with whom he established the Polish Mathematical Society in 1919 and later published the scientific journal ''Studia Mathematica''. In 1920, he received an assistantship at the Lwów Polytechnic, subsequently becoming a professor in 1922 and a member of the Polish Academy of Lear ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
1932
Events January * January 4 – The British authorities in India arrest and intern Mahatma Gandhi and Vallabhbhai Patel. * January 9 – Sakuradamon Incident (1932), Sakuradamon Incident: Korean nationalist Lee Bong-chang fails in his effort to assassinate Emperor Hirohito of Japan. The Kuomintang's official newspaper runs an editorial expressing regret that the attempt failed, which is used by the Japanese as a pretext to attack Shanghai later in the month. * January 22 – The 1932 Salvadoran peasant uprising begins; it is suppressed by the government of Maximiliano Hernández Martínez. * January 24 – Marshal Pietro Badoglio declares the end of Libyan resistance. * January 26 – British submarine aircraft carrier sinks with the loss of all 60 onboard on exercise in Lyme Bay in the English Channel. * January 28 – January 28 incident: Conflict between Japan and China in Shanghai. * January 31 – Japanese warships arrive in Nanking. February * February 2 ** A general ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unbounded Operator
In mathematics, more specifically functional analysis and operator theory, the notion of unbounded operator provides an abstract framework for dealing with differential operators, unbounded observables in quantum mechanics, and other cases. The term "unbounded operator" can be misleading, since * "unbounded" should sometimes be understood as "not necessarily bounded"; * "operator" should be understood as "linear operator" (as in the case of "bounded operator"); * the domain of the operator is a linear subspace, not necessarily the whole space; * this linear subspace is not necessarily closed; often (but not always) it is assumed to be dense; * in the special case of a bounded operator, still, the domain is usually assumed to be the whole space. In contrast to bounded operators, unbounded operators on a given space do not form an algebra, nor even a linear space, because each one is defined on its own domain. The term "operator" often means "bounded linear operator", but in the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuous Dual Space
In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V,'' together with the vector space structure of pointwise addition and scalar multiplication by constants. The dual space as defined above is defined for all vector spaces, and to avoid ambiguity may also be called the . When defined for a topological vector space, there is a subspace of the dual space, corresponding to continuous linear functionals, called the continuous dual space. Dual vector spaces find application in many branches of mathematics that use vector spaces, such as in tensor analysis with finite-dimensional vector spaces. When applied to vector spaces of functions (which are typically infinite-dimensional), dual spaces are used to describe measures, distributions, and Hilbert spaces. Consequently, the dual space is an important concept in functional analysis. Early terms for ''dual'' include ''polarer Raum'' ahn 192 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Open Mapping Theorem (functional Analysis)
In functional analysis, the open mapping theorem, also known as the Banach–Schauder theorem or the Banach theorem (named after Stefan Banach and Juliusz Schauder), is a fundamental result that states that if a bounded or continuous linear operator between Banach spaces is surjective then it is an open map. A special case is also called the bounded inverse theorem (also called inverse mapping theorem or Banach isomorphism theorem), which states that a bijective bounded linear operator T from one Banach space to another has bounded inverse T^. Statement and proof The proof here uses the Baire category theorem, and completeness of both E and F is essential to the theorem. The statement of the theorem is no longer true if either space is assumed to be only a normed vector space; see . The proof is based on the following lemmas, which are also somewhat of independent interest. A linear map f : E \to F between topological vector spaces is said to be nearly open if, for e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
