3 Spaces L p 1. In this part we fix a measure space (, A, µ) (we may assume that µ is complete), and consider the A -measurable functions on it. 2. For f L 1 (, µ) set f 1 = f L 1 = f L 1 (,µ) = f dµ. It follows from the above inequalities that for c C 1 With more work we derive that f + g 1 f 1 + g 1, cf 1 = c f 1. f 1 = 0 f = 0 µ a.e.. Factorising by the subspace of functions equal 0 µ -a.e., or redefining the symbol = between to functions, we conclude that L 1 (, µ) is a normed vector space. 3. Definition of a normed space from B, Ch.2 (B-2). Prove that spaces in B-2, Examples 1, parts 1, 2, 4, 5, 9 (cases l 1, l, and c 0 only ), 13, 14, 15 19 are normed. Sets B(x 0, r), B(x 0, r), and S(x 0, r) in a normed space V. Prove that B(x 0, r) is the closure of B(x 0, r). 4. Complete (Banach) spaces, B-2. Which spaces from the above examples are complete? Provide proofs. 5. We show that L 1 (, µ) is a Banach space. Theorem 1 For any sequence {f j }, f j L 1 (, µ), satisfying there exists f L 1 (, µ) such that f j f k 1 0, j, k, f j f 1 0, j. Proof. 1. First we need to construct f, whose existence is claimed by the theorem. Utilising the Cauchy condition find a subsequence {j n } such that f jn+1 f jn 1 < 2 n. Then for the increasing sequence F N = N f jn+1 f jn 1
we have F N dµ 1 N. Therefore by the monotone convergence theorem f jn+1 f jn = lim N F N = F, F L 1, F dµ 1. This implies in particular, that F < µ -a.e. Consequently the series ( fjn+1 (x) f j n (x) ) converges absolutely for any x \ E with µ(e) = 0. For such x we define def ( f(x) = f n1 (x) + (x) f fjn+1 j n (x) ) = lim N f j N (x). Thus we have constructed f L 1 as the µ -a.e. limit of a certain subsequence {f jn }. 2. Now we show that f f j 1 0 as j. In fact, fix any ε > 0. Find N ε such that f m f n dµ < ε m, n > N ε. Take any n > N ε and test this estimate with m = j N from the previous step. By the Fatou s lemma derive f f n dµ lim inf f jn f n dµ N ε. Since this holds for any n > N ε, the theorem is proved. 6. Let X = (V, ) be a normed space. Prove that X is complete if and only if for any nested sequence of closed balls { B(x n, r n ) } (this means B(x n, r n ) B(x n+1, r n+1 ) for n = 1, 2,... ), such that r n 0 as n, there exists a unique x V for which B(x n, r n ) = {x }. 2
This implication is sometimes called the principle of nested balls. 7. Series in normed and Banach spaces, B-2. Give an example of a convergent series in l 2, which does not converge absolutely (cf. discussion after Theorem 8 in B-2). We saw that the completeness is equivalent to the principle of nested balls. Also the completeness of a normed space is equivalent to the implication Theorem 8, B-2. {absolute convergence} = {convergence}, 8. Dense sets in a normed space, completion of a normed space V, Theorem 7 from B-2. Denote by S the set of all simple functions equipped with the norm 1. Prove that S is a normed space. Prove that L 1 is (isometric to) the completion of S. Prove that in the case of = R n, µ = λ n the space S is not complete. Give an example of (, A, µ) for which S is complete. 9. Jensen s inequality, Theorem 2, B-1, and Young s inequality, Theorem 3, B-1. 10. The Hölder s and Minkowski s discrete inequalities, Theorems 6, 7, B-1, with the Hölder-conjugate exponents 1 p, p (or p and q ) p = p p 1, p = p p 1, 1 p + 1 p = 1. 11. Prove that the sets l p n and l p, 1 p, Examples 6 and 9, Chapter 2, are Banach spaces. Prove that l p1 l p2 if 1 p 1 p 2, and that for any sequence x. x l p 2 x l p 1 12. For any p, 1 p <, define L p = L p (, µ) to be the sets of functions f : C (or f : R ) such that For 1 p < define (i) (ii) f is A measurable; f p dµ <. ( 1/p f p = f L p = f L p (,µ) = f dµ) p. 3
Theorem 2 Let p satisfy 1 < p <. For all f L p, g L p Hölder s inequality holds: fg dµ f p g p. For all u, v L p the Minkovski s inequality holds: the u + v p u p + v p. Prove the theorem following the proofs of the discrete Hölder s and Minkowski s inequalities. For 1 p < use the Minkowski s inequality to prove that (L p, p ) is a normed vector space. 13. Let µ() <. Define the average of f by writing f dµ = 1 f dµ. µ() Prove that L p2 L p1 if 1 p 1 p 2 < (notice, that the inclusions are the opposite compared to the l p -spaces). For that establish the following inequality for the averaged L p -norms: ( 1/p1 ( 1/p2 u dµ) p1 u dµ) p2, p 1 p 2, u L p2. Prove that L p2 L p1, p 1 p 2, in the case µ() =. 14. Adapt the proof of Theorem 1 and establish that L p is a complete normed space if 1 < p <. 15. For an A -measurable function f set ess sup f = inf{c : µ({f > C}) = 0} = inf{c : f(x) C for µ-almost all x}. Hence the more careful notations should be ess supf.,µ Prove that the inf here can be replaced by min, provided it is a finite number. 16. For an A -measurable function f : C define f = f L = f L (,µ) 4 def = ess sup,µ f.
define L = L (, µ) to be the sets of functions f : C (or f : R ) such that (i) f is A measurable; (ii) f <. Prove that L is a normed vector space. 17. Let f, f 1, f 2,... be A -measurable. Prove that f j f 0, j, if and only if there exists E A, µ(e) = 0, such that f j f uniformly on \ E as j. 18. Using the argument from the proof of the previous statement show that L (, µ) is a Banach space. 19. Prove the Hölder s and Minkowski s inequalities for all p, 1 p. Prove that L p2 (, µ) L p1 (, µ) for 1 p 1 p 2 provided µ() <. 20. The space L is in many asects the limit case of L p. For example, prove that if µ() <, then lim f p = f. p 5