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Unformatted text preview: 1 LP spaces
Deﬁnition The Lebesgue space LP (I ) := f : I → R  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 Lp ⊂ Lq ?
1 Exercise Consider f (x) := x1a deﬁned on X := (0, ∞). Formally, x−a dx = 1−a x1−a . ∞ 1 ∞ Recall that for 0 f to exist, 0 f and 1 f must exist. Consider the following cases: I f (x)p dx 1/p <∞ . a = 0: 0 < a < 1: a = 1: a > 1: Proposition (6.9 in [1]) 0 < p < q < r < ∞ =⇒ Lq ⊂ Lp + Lr . 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 (SchroederBernstein, 0.5.21 in [2]) If there are injections f : X → Y , g : Y → X , then there is a bijection h : X → Y . Proof. Tricky. 2 Linear Independence
Deﬁnition {xα }α∈I is linearly independent if each ﬁnite subcollection is linearly independent. Exercise Show that {et , e2t } is linearly independent in (C ([0, 1]), R). Therefore we don’t distinguish between ‘countably inﬁnitedimensional’ and ‘uncountably inﬁnitedimensional’ vector spaces. However function spaces can be extremely weird. Look up ‘Cantor Set’ and ‘Cantor Function’ on Wikipedia. References
[1] G.B. Folland. Real analysis. Wiley New York, 1999. [2] C.C. Pugh. Real mathematical analysis. Springer Verlag, 2002. ...
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This note was uploaded on 04/11/2011 for the course EE 221A taught by Professor Clairetomlin during the Fall '10 term at University of California, Berkeley.
 Fall '10
 ClaireTomlin

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