Operator then we call t a linear functional on v de nition 47 let v be a normed space over f we denote bvf v recall that bvf is the the space of bounded operators for v to f we can v the dual space of v corollary 48 corollary to theorem 45 for any normed space v we have that v is a banach space in the case of p space where p6 1we can now determine exactly what the . The current set of notes is an activity oriented companion to the study of linear functional analysis and operator algebras it is intended as a pedagogical companion for the beginner an introduction to some of the main ideas in this area of analysis a compendium of problems i think are useful in learning the subject and an annotated reading reference list the great majority of the results . 2 contents notations bxy the space of all bounded continuous linear operators from x to y imaget rant the image of a mapping t x y xn w x x n converges weakly to x x the space of all bounded continuous linear functionals on x f or k the scalar fleld which is ror c re im the real and imaginary parts of a complex number. 300 c functional analysis and operator theory a l is bounded if there exists a finite k 0 such that f x klfk k kfk by context klfk denotes the norm of lf in y while kfk denotes the norm of f in x b the operator norm of l is klk sup kfk1 klfk c1 on those occasions where we need to specify the spaces in question we