By Milan Miklavcic

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8 + \bk\2 = - ^ \f(x)\2dx for / 6 L 2 K 4 (2-11) If / e L 2 ( 0 , T T ) , then oo / ( x ) = > J 6jfc sin fcz (series converges in L2(0, ir)) (2-12) k=\ where 2 I"71" bk = f{x) sinfcxdx, k = 1,2,.... 12). 12) is called a Fourier sine series. 6 imply that 2 vl sinkx, k = 1,2,.. 7T form a complete orthonormal set in L 2 (0,7r). 14) is called a Fourier cosine series. (x)=x, = 2xTn{x) - Tn-i(x) for n > 1. 1. ORTHONORMAL SETS 53 and, in particular, |T n (z)| < 1 for n > 0, - 1 < x < 1. Suppose g G C ^ - l j l ] .

Let us show first that T is accretive. Choose any / € D(T) and define A € X* by A(#) = f(xo)g(xo) where XQ G (0,1) is chosen so that h = |/| 2 attains maximum at XQ. A is a normalized tangent functional to / and ReA(27) = a(x 0 )(|/'(xo)| 2 - h"(x0)/2) > 0. 11) we have that R(T — A) = X when |A| is small. 17 are satisfied. 7 Compact Linear Operators We shall now restrict our investigation of operators to a special class which have many of the characteristics of operators in finite dimensional spaces.

If x ^ 0, then 'normalized x' means ||x|| _ 1 x. 1 If H is an inner product space, then for all x and y in H (a) |(x,y)| < ||x||||y|| (Schwarz Inequality) (b) ||* + y | | < | | * | | + ||y||. 47 CHAPTER 2. LINEAR OPERATORS 48 PROOF in IN HUBERT SPACES (a) is obvious if x = 0; otherwise it follows by taking 8 = — (re, y)/||x|| 0 < \\6x + yf = |«5|2||x||2 + 2Re(fe, 2/ ) + ||„|| 2 . This identity, with 6 = 1 and (a), implies (b). 1) defines a norm on an inner product space H. If this norm makes H complete, then H is said to be a Hilbert space.