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Exercise 2.8. Let p �= q; consider the norms of �p and �q on the linear space f. Then these norms are not equivalent. Hint: use the sequence gn as above. pr p∞ Exercise2.11.Provethatifp<r,then� ⊂�. Hint: firstprovethat� ⊂� . Next, given x ∈ �p, renormalise so that �x�∞ � 1; now use the inequality |x|r � |x|p if |x| � 1. Exercise 2.26. 1. Let p � 1. Consider the function f (t) = t − (tp/p), t � 0. Prove that f (t) �f(1)forallt�0. 2. Substitute t = |xi|/|yi|q−1; rearrange to obtain |xiyi| � p1|xi|p + 1q|yi|q. (Hint: remember to use p1 + 1q = 1). 3. Take a sum over i and use an additional parameter a > 0 to obtain i=1 �n 1 1 |x y | � ap�x�p + a−q�y�q. iippqq 4. Choose an appropriate value of a to obtain the H ̈older inequality. (Hint: minimize the r.h.s. in a > 0). Exercise 2.30. Consider the sequence xn = √ 1 , n � 1. For which p � 1 is the n log(1+n) statement x ∈ �p true? Exercise 2.32. Prove that the set {x ∈ � | |x1| < n=2|xn|2 } is open in � . Hint: if f : �1 → R is a continuous function, then the pre-image f−1((0,∞)) is open. Exercise 2.33. Prove that the set {x ∈ �2 | |x2n| = 1/n ∀n ∈ N} is closed in �2. �∞ 1 −n 1 3 Lebesgue integration and Lp spaces 3.1 Riemann integral This subsection recalls some material you saw in Analysis I. Fix [a, b]; we discuss functions f : [a, b] → C. A step function is a finite linear combina- tion of characteristic functions of bounded intervals (the intervals can be open, closed, or neither): For a step function f, the integral is defined in an obvious way: a n � f(x) = cnχδn(x). n (3.1) (3.2) �b� f(x)dx = cnμ(δn), where μ(δn) is the length of the interval δn. Let R[a,b] denote the Riemann integrable functions on [a, b]. In a certain sense, these are the functions that can be well-approximated by step functions, and we extend the definition of the integral to all such functions. Let B[a, b] be the set of all bounded functions on [a, b] with the norm �f�∞ = sup |f(x)|. x∈[a,b] The integral can be viewed as a linear functional I : R[a, b] → C with the property that Recall that C[a, b] ⊂ R[a, b] ⊂ B[a, b]. |I(f)| � (b − a)�f�∞. Thus, |I(f1 −f2)| � (b−a)�f1 −f2�∞ and so I is a continuous functional on R[a,b], viewed as a subspace of B[a, b]. Exercise 3.15. Prove that this happens if and only if the integral �R|f(x)|dμ(x) is finite. Denote the class of all such functions by L1(R). For f ∈ L1(R), define ���� fdμ = f+dμ − f−dμ. RRR The class L1(a,b) and the integral ab can be defined in a similar way. Exercise 3.32. (i) Give an example of a sequence of functions fn ∈ L1(R) ∩ L2(R) such that fn → 0 in L1(R) but fn �→ 0 in L2(R). (ii) Give an example of a sequence of functions gn ∈ L1(R) ∩ L2(R) such that gn → 0 in L2(R) but gn �→ 0 in L1(R). Exercsie 3.36. Let f ∈ L1(R). For t ∈ R, let ft be the function ft(x) = f(x − t). Prove thatf f inL1(R)ast 0. t→→ Hint: approximate f by suitable functions. Exercise 3.38. Prove that if f ∈ L1(R) is uniformly continuous, then limx→±∞ f(x) = 0. Hint: argue by contradiction. 2π Exercise 4.44. Prove Proposition 4.23 Exercise 4.47. Let M be a subspace of a Hilbert space H and suppose that T : M → C is a bounded linear functional. Show that there exists a bounded linear functional S : H → C such that S = T on M and �S� = �T� (cf. the Hahn-Banach theorem in Section 5.2). Exercise 4.48. Let X be an inner product space and let {x1, . . . , xn} be an orthonormal set in X. Prove that � �n � �x − ckxk� � k=1 � is minimized by choosing ck = �x, xk �. Exercise 4.49. Show that if a Hilbert space contains an infinite orthonormal system then √2 the unit ball contains infinitely many disjoint translates of a ball of radius 4 . 1 ∞�∞−n ∞∗ Exercise5.24.Forx∈� ,letf(x)= n=1xn2 .Determinethenormoffin(� );in (�1)∗; in any (�p)∗, 1 < p < ∞. Exercise 5.25. Let X be a reflexive Banach space. Using Corollary 5.14, prove that for any�∈X∗ thereisanelementx∈X suchthat�x�X =1and�(x)=���X∗. Exercise 5.26. Let � : C[−1, 1] → C be defined by �0 �1 �(f) = f(x)dx − f(x)dx. −1 0 Prove that � is a bounded linear functional on C[−1,1] and determine its norm ���C∗. Prove that there is no element f ∈ C[−1,1] such that �f�C = 1 and �(f) = ���C ∗ . (Hint: Assume such f exists. By multiplying by an appropriately chosen unimodular complex number and by taking real parts, reduce the problem to the case when f is real- valued. Now consider 3 possibilities: f(0) = 1, f(0) = −1, |f(0)| < 1. In each case, prove that |�(f)| < ���C Compare this with the result of the previous problem; what can be said about the space C[−1, 1]? Exercise 5.27. Using the Hahn-Banach Theorem, prove that there exists a linear func- tional λ on �∞ with the following property. If x ∈ �∞ is such that the limit x∞ = limn→∞ xn exists, then λ(x) = x∞. That is, λ extends the definition of the limit to all bounded se- quences. Exercise 5.29. Let � : L3[0, 1] → C be th�e linear functional 1√ �(f) = f( x)dx. 0 Find the norm of � (in the dual space to L3). Exercise5.30.LetXbeaBanachspace.For�∈X∗,letKer�={x∈X:�(x)=0}. 1.Let�∈X∗ begiven,andletx0 ∈X\Ker�. Showthatanyx∈Xhasaunique representation as x = αx0 + y where α ∈ C, y ∈ Ker �. 2. Let �1, �2 ∈ X∗. Show that if Ker �1 = Ker �2, then �1 and �2 are proportional. 3. Show that ���X∗ can be interpreted geometrically as the reciprocal of the distance from the hyperplane �−1(1) to the origin. ∗ .) Exercise 6.24. For the following subsets of C [0, 1], decide whether these subsets are com- pact: 1. The whole space C[0, 1]; 2. The unit ball in C [0, 1]; 3. The unit sphere: {f ∈ C[0,1] | �f�C = 1}; 4. The set of all polynomials; 5. The set of all polynomials whose coefficients are bounded above by 1. 6. The set of all polynomials of degree � n whose coefficients ai satisfy |ai| � 1. Exercise 6.28. For each of the following functions f, decide whether f is uniformly con-tinuous on the given set and prove your answer. 1. f(x)=sin(1/x),x∈(0,1). 2. f(x) = x sin(1/x), x ∈ (0, 1). 3. f(x) = √x sin(1/x), x ∈ (0, 1). 4. f(x)=1/(1+x2(sinx)2),x∈R. 5. f(x) = (sin(x2))/(1 + x2), x ∈ R. Exercises Exercise 7.13. Fix a sequence a ∈ � . Consider the set K ⊂ � of all sequences x satisfying | x ||a�|,n∈N.ProvethatKiscompactin�2. nn 22 Exercise 7.14. Consider the set �21 of all sequences x satisfying �∞n=1 n2|xn|2 < ∞. Prove that �21 is a normed linear space with the norm ��∞ 2 2�1/2 �x� = n |xn| . n=1 Prove that the embedding �21 ⊂ �2 is compact. Exercise 7.15. For each of the following sets of functions, decide whether this set is totally bounded in the given space and prove your answer. You may use the Arzela-Ascoli theorem. 1. fa(x) = sin(ax), a ∈ [0, 1], in C[0, π]. 2. fa(x) = sin(ax), a ∈ [0, ∞), in C[0, π]. 3. fa(x) = xa, a ∈ [0,2], in C[0,1]. 4. fa(x) = xa, a ∈ [1,∞), in C[0,1]. 5. fa(x) = xa, a ∈ [1,∞), in C[0, 12].

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