Closures and Adjoints of Linear Differential Operators
Israel Halperin · Annals of Mathematics · 1937
In a real or complex Banach space B of elements f, g, *. , with norm IIf II, 1 g 91, * .. , an operator T is a function whose values lie in the same Banach space but which need not be everywhere defined nor single-valued.' T2 is said to be an extension of T1, ,in symbols T2 D T1, or T1 C T2, if for everyf, the set of values of T2f includes all values of Tif. The operator T is called linear if the set of values of T(cf + dg) includes all values of cTf + dTg, for all f, g and all numbers c, d (real or complex, depending on B). A closure operator T (for arbitrary T) is defined by the condition that Tf has g as one of its values if and only if