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Continuous linear operator information


In functional analysis and related areas of mathematics, a continuous linear operator or continuous linear mapping is a continuous linear transformation between topological vector spaces.

An operator between two normed spaces is a bounded linear operator if and only if it is a continuous linear operator.

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Continuous linear operator

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mathematics, a continuous linear operator or continuous linear mapping is a continuous linear transformation between topological vector spaces. An operator between...

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Bounded operator

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In functional analysis and operator theory, a bounded linear operator is a linear transformation L : X → Y {\displaystyle L:X\to Y} between topological...

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Compact operator

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In functional analysis, a branch of mathematics, a compact operator is a linear operator T : X → Y {\displaystyle T:X\to Y} , where X , Y {\displaystyle...

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Operator norm

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mathematics, the operator norm measures the "size" of certain linear operators by assigning each a real number called its operator norm. Formally, it...

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Integral linear operator

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topological vector spaces (TVSs) X and Y. An integral linear operator is a continuous linear operator that arises in a canonical way from an integral bilinear...

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Continuous linear extension

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Theorems connecting continuity to closure of graphs Continuous linear operator Densely defined operator – Function that is defined almost everywhere (mathematics)...

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Operator algebra

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functional analysis, a branch of mathematics, an operator algebra is an algebra of continuous linear operators on a topological vector space, with the multiplication...

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Linear map

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may be continuous. If its domain and codomain are the same, it will then be a continuous linear operator. A linear operator on a normed linear space is...

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Normal operator

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especially functional analysis, a normal operator on a complex Hilbert space H is a continuous linear operator N : H → H that commutes with its Hermitian...

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Hermitian adjoint

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specifically in operator theory, each linear operator A {\displaystyle A} on an inner product space defines a Hermitian adjoint (or adjoint) operator A ∗ {\displaystyle...

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Convolution

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invariant continuous linear operator on L1 is the convolution with a finite Borel measure. More generally, every continuous translation invariant continuous linear...

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Operator topologies

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bounded linear operators on a Banach space X. Let ( T n ) n ∈ N {\displaystyle (T_{n})_{n\in \mathbb {N} }} be a sequence of linear operators on the Banach...

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Functional analysis

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are the continuous linear operators defined on Banach and Hilbert spaces. These lead naturally to the definition of C*-algebras and other operator algebras...

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Dual space

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is a subspace of the dual space, corresponding to continuous linear functionals, called the continuous dual space. Dual vector spaces find application in...

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Unbounded operator

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This is a linear operator, since a linear combination a f  + bg of two continuously differentiable functions  f , g is also continuously differentiable...

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Discontinuous linear map

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{\displaystyle f} is a bounded linear operator and so is continuous. In fact, to see this, simply note that f is linear, and therefore ‖ f ( x ) − f (...

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Densely defined operator

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function. In a topological sense, it is a linear operator that is defined "almost everywhere". Densely defined operators often arise in functional analysis as...

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Operator theory

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mathematics, operator theory is the study of linear operators on function spaces, beginning with differential operators and integral operators. The operators may...

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Transpose of a linear map

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Y} is a weakly continuous linear operator between topological vector spaces X {\displaystyle X} and Y {\displaystyle Y} with continuous dual spaces X ′...

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Nuclear operator

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nuclear operators are an important class of linear operators introduced by Alexander Grothendieck in his doctoral dissertation. Nuclear operators are intimately...

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Bochner integral

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functions. If T : B → B ′ {\displaystyle T\colon B\to B'} is a continuous linear operator between Banach spaces B {\displaystyle B} and B ′ {\displaystyle...

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Closed graph property

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discontinuous linear operator. Every continuous linear operator valued in a Hausdorff topological vector space (TVS) has a closed graph and recall that a linear operator...

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Operator ideal

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of mathematics, an operator ideal is a special kind of class of continuous linear operators between Banach spaces. If an operator T {\displaystyle T}...

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Banach space

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Every continuous linear operator is a bounded linear operator and if dealing only with normed spaces then the converse is also true. That is, a linear operator...

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Uniform boundedness principle

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continuous linear operators from X {\displaystyle X} into Y {\displaystyle Y} . Suppose that F {\displaystyle F} is a collection of continuous linear...

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