Integration: Exercises R Ch 4

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last updates: 10 April 2011

Integration: Exercises R Ch 4

  1. If M is a closed subspace of H, prove that M= (M) . Is there a similar true statement for spaces M which are not necessarily closed?
  2. Let {xn | n=1,2,} be a linearly independent set of vectors in H. Show that the following construction yields an orthonormal set {un} such that {x1, xN} and {u1, uN} have the same span for all N.
    Put u1 =x1/ x1. Having u1, un-1 define
    vn = xn - =1 n-1 (xn, ui) ui,      un= vn/ vn.
    Note that this leads to a proof of the existence of a maximal orthonormal set in separable Hilbert spaces which makes no appeal to the Hausdorff maximality principle. (A space is separable if it contains a countable dense subset.)
  3. Show that Lp(T) is separable if 1p<, but that L(T) is not separable.
  4. Show that H is separable if and only if H contains a maximal orthonormal system which is at most countable.
  5. If M={ x| Lx=0} , where L is a continuous linear functional on H, prove that M is a space of dimension 1 (unless M=H).
  6. Let {un} be an orthonormal set in H. Show that this gives an example of a closed and bounded set which is not compact. Let Q be the set of all xH of the form
    x = i=1 cnun      ( where |cn| 1n ).
    Prove that Q is compact. (Q is called the Hilbert cube.)
    More generally, let {δn} be a sequence of positive numbers, and let S be the set of all x H of the form
    x = i=1 cnun      ( where |cn| δn ).
    Prove that S is compact if and only if 1 δn2 <.
    Prove that H is not locally compact.
  7. Suppose {an} is a sequence of positive numbers such that an bn<, whenever bn0 and bn2 <. Prove that an2 <.
    Suggestion: If an2 = then there are disjoint sets Ek (k=1,2,3,) so that
    nEk an2 >1.
    Define bn so that bn= ckan for nEk. For suitably chosen ck, an bn = although bn2 <.
  8. If H1 and H2 are two Hilbert spaces, prove that one of them is isomorphic to a subspace of the other. (Note that every closed subspace of a Hilbert space is a Hilbert space.)
  9. If A[0,2π] and A is measurable, prove that
    limn A cosnx dx = limn A sinnx dx = 0 .
  10. Let n1< n2< n3< be positive integers, and let E be the set of all x[0,2π] at which {sinnkx} converges. Prove that m(E)=0. Hint: 2sin2α = 1-cos2α, so sinnkx ±1/2 a.e. on E, by Exercise 9.
  11. Find a nonempty closed set E in L2 (T) that contains no element of smallest norm.
  12. The constants ck in Sec. 4.24 were shown to be such that k-1ck is bounded. Estimate the relevant integral more precisely and show that
    0< limk k-1/2 ck < .
  13. Suppose f is a continuous function on 1 with period 1. Prove that
    0< limN 1N n=1 N f(nα) = 01 f(t) dt
    for every irrational real number α. Hint: Do it first for
    f(t) = exp(2πikt), k=0,±1, ±2, .
  14. Compute
    mina,b,c -11 | x3-a-bx -cx2| 2 dx
    and find
    max -11 x3 g(x) dx
    where g is subject to the restrictions
    -11 g(x) dx = -11 xg(x) dx = -11 x2g(x) dx = 0; -11 |g(x)| 2 dx = 1 .
  15. Compute
    mina,b,c 0 | x3-a-bx -cx2| 2 e-x dx .
    State and solve the corresponding maximum problem, as in Exercise 14.
  16. If x0H and M is a closed linear subspace of H, prove that
    min { x-x0 | xM } = max { |(x0,y) | | yM, y =1 } .
  17. Show that there is a continuous one-to-one mapping γ of [0,1] into H such that γ(b) -γ(a) is orthogonal to γ(d) -γ(c) whenever 0abc d1. (γ may be called a "curve with orthogonal increments.") Hint: Take H=L2, and consider characteristic functions of certain subsets of [0,1].
  18. Define us(t) =eist for all s1, t1. Let X be the complex vector space consisting of all finite linear combinations of these functions us. If fX and gX, show that
    (f,g) = limA 12A -AA f(t) g(t) dt
    exists. Show that this inner product makes X into a unitary space whose completion is a non-separable HIlbert space H. Show also that {us | s1} is a maximal orthonormal set in H.
  19. Fix a positive integer N. Put ω=e2πi /N. Prove the orthogonality relations
    1N n=1N ωnk = { 1, if k=0, 0, if 1kN-1,
    and use them to derive the identities
    (x,y) = 1N n=1N x+ωny 2 ωn
    that hold in every inner product space if N3. Show also that
    (x,y) = 12π -ππ x+eiθy 2 eiθ dθ .

Notes and References

These exercises are taken from [Ru, Chapt. 4] for a course in "Measure Theory" at the Masters level at University of Melbourne.


[RuB] W. Rudin, Principles of Mathematical Analysis, Third edition, McGraw-Hill, 1976. MR??????.

[Ru] W. Rudin, Real and complex analysis, Third edition, McGraw-Hill, 1987. MR0924157.

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