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**Sample text**

21 Theorem. Let f E L1(R). Then (a) F{f(xn = F{J(-xn. (b) F{f(x - yn = F{f(xne- iky . a> O. (c) F{J(axn = (l/a)F{J(x/an, Theorem. If f is a continuous piecewise differentiable function, limlxl-+oo f(x) = 0, then F{f'} = -ikF{J}. f, I' E L 1 (R), and To prove this theorem we apply integration by parts. Corollary. If f is a continuous piecewise n-times differentiable function with f, ... , f(n) E L 1 (R), and limlxl-+oo f(k)x = 0 for k = 0, ... , n -1, then 1', Theorem. (Convolution Theorem). Let f, 9 E L1 (R).

The operator A is termed self-adjoint if A* = A or, equivalently, if (J, Ag) = (Af, g) for all f, 9 E 1l. Example. Let 1l = C 2 • Then A = (~i ~) is a self-adjoint operator (hermitian matrix). In the case where A is an unbounded operator in 1l we again define its adjoint, A * , by the same formula, except that f is confined to D(A) and 9 to the domain D(A*), which is specified as follows: 9 belongs to D(A*) if there is a vector gA in 1l such that (J, gA) = (Af, g) for all f in D(A) in which case gA = A*g.

Example. Let B be a bounded operator. We define A := B* B. Then A is a bounded self-adjoint operator and the operator A is positive. " Remark. If B is unbounded, then B* B need not be self-adjoint. Remark. An operator product AB is defined on a domain V(AB) = { v E V(B) : Bv E V(A) } and then (AB)v := A(Bv). Therefore V(A*A) may be smaller than V(A). Next we summarize the algebraic properties of the operator norm. It follows from the definitions of the norm and the adjoint of a bounded operator, together with the triangular inequality that if A, B are bounded operators and c E C, then IlcA11 IclllAIl IIAI12 IIA*AII IIA+BII < IIAII+IIBII IIABII < IIAIIIIBII· Definition.