In functional analysis, a branch of mathematics, a closed linear operator or often closed operator is a linear operator whose graph is closed. It is a basic example of an unbounded operator.
The closed graph theorem says a closed linear operator between Banach spaces is continuous; thus is a bounded operator. Hence, a closed linear operator that is used in practice is typically not defined on a Banach space.
We assume that and are topological vector spaces, such as Banach spaces for example, and that Cartesian products, such as are endowed with the product topology. The graph of this function is the subset of where denotes the function's domain. The map is said to have a closed graph (in ) if its graph is a closed subset of product space (with the usual product topology). Similarly, is said to have a sequentially closed graph if is a sequentially closed subset of
A closed linear operator is a linear map whose graph is closed (it need not be continuous or bounded). It is common in functional analysis to call such maps "closed", but this should not be confused the non-equivalent notion of a " closed map" that appears in general topology.
Partial functions
It is common in functional analysis to consider partial functions, which are functions defined on a dense subset of some space A partial function is declared with the notation which indicates that has prototype (that is, its domain is and its codomain is ) and that is a dense subset of Since the domain is denoted by it is not always necessary to assign a symbol (such as ) to a partial function's domain, in which case the notation or may be used to indicate that is a partial function with codomain whose domain is a dense subset of [1] A densely defined linear operator between vector spaces is a partial function whose domain is a dense vector subspace of a TVS such that is a linear map. A prototypical example of a partial function is the derivative operator, which is only defined on the space of once continuously differentiable functions, a dense subset of the space of continuous functions.
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 function is (as before) the set However, one exception to this is the definition of "closed graph". A partial function is said to have a closed graph (respectively, a sequentially closed graph) if is a closed (respectively, sequentially closed) subset of in the product topology; importantly, note that the product space is and not as it was defined above for ordinary functions. [note 1]
A linear operator is closable in if there exists a vector subspace containing and a function (resp. multifunction) whose graph is equal to the closure of the set in Such an is called a closure of in , is denoted by and necessarily extends
If is a closable linear operator then a core or an essential domain of is a subset such that the closure in of the graph of the restriction of to is equal to the closure of the graph of in (i.e. the closure of in is equal to the closure of in ).
Suppose is a linear operator between Banach spaces.
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In functional analysis, a branch of mathematics, a closed linear operator or often closed operator is a linear operator whose graph is closed. It is a basic example of an unbounded operator.
The closed graph theorem says a closed linear operator between Banach spaces is continuous; thus is a bounded operator. Hence, a closed linear operator that is used in practice is typically not defined on a Banach space.
We assume that and are topological vector spaces, such as Banach spaces for example, and that Cartesian products, such as are endowed with the product topology. The graph of this function is the subset of where denotes the function's domain. The map is said to have a closed graph (in ) if its graph is a closed subset of product space (with the usual product topology). Similarly, is said to have a sequentially closed graph if is a sequentially closed subset of
A closed linear operator is a linear map whose graph is closed (it need not be continuous or bounded). It is common in functional analysis to call such maps "closed", but this should not be confused the non-equivalent notion of a " closed map" that appears in general topology.
Partial functions
It is common in functional analysis to consider partial functions, which are functions defined on a dense subset of some space A partial function is declared with the notation which indicates that has prototype (that is, its domain is and its codomain is ) and that is a dense subset of Since the domain is denoted by it is not always necessary to assign a symbol (such as ) to a partial function's domain, in which case the notation or may be used to indicate that is a partial function with codomain whose domain is a dense subset of [1] A densely defined linear operator between vector spaces is a partial function whose domain is a dense vector subspace of a TVS such that is a linear map. A prototypical example of a partial function is the derivative operator, which is only defined on the space of once continuously differentiable functions, a dense subset of the space of continuous functions.
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 function is (as before) the set However, one exception to this is the definition of "closed graph". A partial function is said to have a closed graph (respectively, a sequentially closed graph) if is a closed (respectively, sequentially closed) subset of in the product topology; importantly, note that the product space is and not as it was defined above for ordinary functions. [note 1]
A linear operator is closable in if there exists a vector subspace containing and a function (resp. multifunction) whose graph is equal to the closure of the set in Such an is called a closure of in , is denoted by and necessarily extends
If is a closable linear operator then a core or an essential domain of is a subset such that the closure in of the graph of the restriction of to is equal to the closure of the graph of in (i.e. the closure of in is equal to the closure of in ).
Suppose is a linear operator between Banach spaces.
Cite error: There are <ref group=note>
tags on this page, but the references will not show without a {{reflist|group=note}}
template (see the
help page).