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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 μ on a vector x to be linear in both μ and x . Usually, we use the ordinary notation μ ( 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, μ ) for the action of μ on x . Furthermore, I will take this notation to be linear in both μ and x (note that in most of the lecture notes, I take this notation to be linear in x and antilinear in μ ). 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→ ˆ g be the canonical embedding of L p ( E ) into L p ( E ) ∗∗ . That is, ˆ g : L p ( E ) ∗ → F is given by ( μ, ˆ g ) = ( g, μ ) , μ ∈ L p ( E ) ∗ ....
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- Summer '08
- Vector Space, LP, Linear map, measure, Functional, Topological vector space