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# rec2 - 1 LP spaces Denition The Lebesgue space LP(I:= f I R...

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1 L P spaces Definition The Lebesgue space L P ( I ) := n f : I R | (R I | f ( x ) | p dx ) 1 /p < o . Exercise Naturally a vector space under the operations ( f + g )( x ) := f ( x ) + g ( x ) , ( λf )( x ) := λf ( x ) . (Hint: you need the Minkowski Inequality to prove this.) Question L p L q ? Exercise Consider f ( x ) := 1 x a defined on X := (0 , ). Formally, R x - a dx = 1 1 - a x 1 - a . Recall that for R 0 f to exist, R 1 0 f and R 1 f must exist. Consider the following cases: a = 0: 0 < a < 1: a = 1: a > 1: Proposition (6.9 in [1]) 0 < p < q < r < = L q L p + L r . Proof. Requires measure theory. At Berkeley, take Math 202. Cardinality Definition If f : X Y is bijective then X and Y have equal cardinality . Definition If f : X N is injective then X is countable ; otherwise uncountable . Exercise Z is countable. Exercise Q is countable. Exercise R is uncountable. Theorem ( Schroeder-Bernstein , 0.5.21 in [2]) If there are injections f : X Y , g : Y X , then there is a bijection

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