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Unformatted text preview: C Functional Analysis and Operator Theory C.1 Linear Operators on Normed Spaces In this section we will review the basic properties of linear operators on normed spaces. Definition C.1 (Notation for Operators). Let X, Y be vector spaces, and let T : X Y be a function mapping X into Y. We write either T ( f ) or Tf to denote the image under T of an element f X. (a) T is linear if T ( f + g ) = T ( f )+ T ( g ) for every f,g X and , C . (b) T is antilinear if T ( f + g ) = T ( f )+ T ( g ) for f,g X and , C . (c) T is injective if T ( f ) = T ( g ) implies f = g. (d) The kernel or nullspace of T is ker( T ) = { f X : T ( f ) = 0 } . (e) The range of T is range( T ) = { T ( f ) : f X } . (f) The rank of T is the vector space dimension of its range, i.e., rank( T ) = dim(range( T )) . In particular, T is finiterank if range( T ) is finitedimen sional. (g) T is surjective if range( T ) = Y. (h) T is a bijection if it is both injective and surjective. We use either the symbol I or I X to denote the identity map of a space X onto itself. A mapping between vector spaces is often referred to as an operator or a transformation , especially if it is linear. We introduce the following terminol ogy for operators on normed spaces. Definition C.2 (Operators on Normed Spaces). Let X, Y be normed linear spaces, and let L : X Y be a linear operator. 300 C Functional Analysis and Operator Theory (a) L is bounded if there exists a finite K 0 such that f X, bardbl Lf bardbl K bardbl f bardbl . By context, bardbl Lf bardbl denotes the norm of Lf in Y, while bardbl f bardbl denotes the norm of f in X. (b) The operator norm of L is bardbl L bardbl = sup bardbl f bardbl =1 bardbl Lf bardbl . (C.1) On those occasions where we need to specify the spaces in question, we will write bardbl L bardbl X Y for the operator norm of L : X Y. (c) We set B ( X,Y ) = braceleftbig L : X Y : L is bounded and linear bracerightbig . If X = Y then we write B ( X ) = B ( X,X ) . (f) If Y = C then we say that L is a functional . The set of all bounded linear functionals on X is the dual space of X, and is denoted X = B ( X, C ) = braceleftbig L : X C : L is bounded and linear bracerightbig . Another common notation for the dual space is X . Notation C.3 (Terminology for Unbounded Operators). Unbounded operators are often not defined on the entire space X but only on some dense subspace. For example, the differentiation operator Df = f is not defined on all of L p ( R ) , but it is common to refer to the differentiation operator D on L p ( R ), with the understanding that D is only defined on some associated dense subspace such as L p ( R ) C 1 ( R ) or S ( R ) . Another common terminology is to write that D : L p ( R ) L p ( R ) is densely defined, again meaning that the domain of D is a dense subspace of L p ( R ) and D maps this domain into L p ( R ) ....
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 Fall '09
 HEIL
 Vector Space

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