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 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 dom(T) of X and takes values in Y, written T : dom(T) ⊆ XY. Sometimes this is abbreviated as T : XY when the context makes it clear that X might not be the set-theoretic domain of T.


is a densely defined operator from C0([0, 1]; R) to itself, defined on the dense subspace C1([0, 1]; R). Note also that the operator D is an example of an unbounded linear operator, since
This unboundedness causes problems if one wishes to somehow continuously extend the differentiation operator D to the whole of C0([0, 1]; R).


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