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Unformatted text preview: FUNCTIONAL ANALYSIS LECTURE NOTES: REFLEXIVITY OF L p CHRISTOPHER HEIL Notation: When talking about reflexivity, it is convenient to take the action of a func- tional on a vector x to be linear in both and x . Usually, we use the ordinary notation ( x ) to denote an action that is linear in both variables. However, in this note I will use a slightly different notation and write ( x, ) for the action of on x . Furthermore, I will take this notation to be linear in both and x (note that in most of the lecture notes, I take this notation to be linear in x and antilinear in ). Theorem 1. If E R is Lebesgue measurable, then L p ( E ) is reflexive for each 1 < p < . Proof. Let p be the dual index. Let L : L p ( E ) L p ( E ) g mapsto g be the canonical embedding of L p ( E ) into L p ( E ) . That is, g : L p ( E ) F is given by ( , g ) = ( g, ) , L p ( E ) ....
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This note was uploaded on 08/25/2011 for the course MATH 6338 taught by Professor Staff during the Summer '08 term at Georgia Institute of Technology.
- Summer '08