Integration: Exercises R Ch 5

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

Last updates: 9 April 2011

Integration: Exercises R Ch 5

  1. Let X consist of two points a and b, put μ({a}) = μ({b}) = 12 , and let Lp(μ) be the resulting real Lp-space. Identify each real function f on X with the point (f(a), f(b)) in the plane, and sketch the unit balls of Lp(μ), for 0<p. Note that they are convex if and only if 1p. For which p is this unit ball a square? A circle? If μ({a}) μ({b}) , how does this situation differ from the preceding one?
  2. Prove that the unit ball (open or closed) is convex in every normed linear space.
  3. If 1<p< prove that the unit ball of Lp(μ) is strictly convex; this means that if
    fp = gp =1,      fg,      h= 12(f+g),
    then hp <1. (Geometrically, the surface of the ball contains no straight lines.) Show that this fails in every L1(μ), in every L(μ), and in every C(X). (Ignore trivialities, such as spaces consisting of only one point.)
  4. Let C be the space of all continuous functions on [0,1], with the supremum norm. Let M consist of all fC for which
    01/2 f(t) dt - 1/21 =1.
    Prove that M is a closed convex subset of C which contains no element of minimal norm.
  5. Let M be the set of all f L1([0,1]), relative to Lebesgue measure, such that
    01 =1.
    Show that M is a closed convex subset of L1([0,1]) which contains infinitely many elements of minimal norm. (Compare this and Exercise 4 with Theorem 4.10.)
  6. Let f be a bounded linear functional on a subspace M of a Hilbert space H. Prove that f has a unique norm-preserving extension to a bounded linear functional on H, and that this extension vanishes on M.
  7. Construct a bounded linear functional on some subspace of L1(μ) which has two (hence infinitely many) distinct norm-preserving linear extensions to L1(μ).
  8. Let X be a normed linear space, and let X* be its dual space with the norm
    f = sup{ |f(x)| | x1}.
    (a)   Prove that X* is a Banach space.
    (b)   Prove that the mapping ff(x) is, for each xX, a bounded linear functional on X*, of norm x. (This gives a natural imbedding of X in its "second dual" X**, the dual space of X*.)
    (c)   Prove that {x} is bounded if {xn} is a sequence in X such that {f(xn)} is bounded for every fX*.
  9. Let c0, 1, and be the Banach spaces consisting of all complex sequences x= {ξi}, i=1,2,, defined as follows:
    x1 if and only if x1 = |ξi| <,
    x if and only if x =sup| ξi|<,
    c0 is the subspace of consisting of all x for which ξi0 as i.
    Prove the following four statements.
    (a)   If y= {ηi} 1 and Λx= ξi ηi for every xc0, then Λ is a bounded linear functional on c0 and Λ= y1. Moreover, every Λ (c0)* is obtained this way. In brief, (c0)* =1.
    (b)   In the same sense, (1)* =.
    (c)   Every y1 induces a bounded linear functional on , as in (a). However, this does not give all of () *, since () * contains nontrivial functionals that vanish on all of c0.
    (d)   c0 and 1 are separable but is not.
  10. If αi ηi converges for every sequence {ξi} such that ξi0 as i, prove that |αi| <.
  11. For 0<α1, let Lipα denote the space of all complex functions f on [a,b] for which
    Mf= supst |f(s) -f(t)| |s-t| α < .
    Prove that Lipα is a Banach space, if f = |f(a)| +Mf; also if
    f = Mf+ supx |f(x)| .
    (The members of Lipα are said to satisfy a Lipschitz condition of order α.)
  12. Let K be a triangle (two dimensional figure) in the plane, let H be the set consisting of the vertices of K, and let A be the set of all real functions f on K, of the form
    f(xy) = αx+βy+ γ      (α,β, and γ real).
    Show that to each (x0, y0)K there corresponds a unique measure μ on H such that
    f(x0, y0) = H fdμ .
    (Compare Sec. 5.22.)
    Replace K by a square, let H again be the set of its vertices, and let A be as above. Show that to each point of K there still corresponds a measure on H, with the above property, but that uniqueness is now lost.
    Can you extrapolate to a more general theorem? (Think of other figures, higher dimensional spaces.)
  13. Let {fn} be a sequence of continuous complex functions on a (nonempty) complete metric space X, such that f(x) = limfn exists (as a complex number) for every xX.
    (a)   Prove that there is an open set V and an number M< such that |fn(x)| <M for all xV and for n=1,2,3,.
    (b)   If ϵ>0, prove that there is an open set V and an integer N such that |f(x) -fn(x)| ϵ if xV and nN.
    Hint for (b): For N=1,2,3,, put
    AN = {x | | fm(x) - fn(x) | ϵ if mN and nN } .
    Since X= AN, some AN has a nonempty interior.
  14. Let C be the space of all real continuous functions on I= [0,1] with the supremum norm. Let Xn be the subset of C consisting of those f for which there exists a tI such that | f(s) - f(t) | n|s-t| for all sI. Fix n and prove that each open set in C contains an open set which does not intersect Xn. (Each fC can be uniformly approximated by a zigzag function g with very large slopes, and if g-h is small, hXn.) Show that this implies the existence of a dense Gδ in C which consists entirely of nowhere differentiable functions.
  15. Let A= (aij) be an infinite matrix with complex entries, where i,j =0,1,2,. A associates with each sequence {sj} a sequence {σi}, defined by
    σi = j=0 aij sj      (i= 1,2,3,),
    provided that these series converge.
    Prove that A transforms every convergent sequence {sj} to a sequence {σi} which converges to the same limit if and only if the following conditions are satisfied:
    limi aij =0 ,      for each j. (a)
    supi j=0 |aij| < . (b)
    limi j=0 aij =1 . (c)
    The process of passing from {sj} to {σi} is called a summability method. Two examples are
    aij = { 1i+1, if 0ji, 0, if i<j,
    aij = (1-ri) rij, 0<ri<1, ri1.
    Prove that each of these also transforms some divergent sequences {sj} (even some unbounded ones) to convergent sequences {σi}.
  16. Suppose X and Y are Banach spaces, and suppose Λ is a linear mapping of X into Y, with the following property: For every sequence {xn} in X for which x=limxn and y=limΛxn exist, if is true that y=Λx. Prove that Λ is continuous.
    This is the so called "closed graph theorem". Hint: Let XY be the set of all ordered pairs (x,y), xX and yY, with addition and scalar multiplication defined componentwise. Prove that XY is a Banach space, if (x,y) =x+ y . The graph G of Λ is the subset of XY formed by the pairs (x,Λx), xX. Note that our hypothesis says that G is closed; hence G is a Banach space. Note that (x,Λx) x is continuous, one-to-one, and linear and maps G onto X.
    Observe that there exist nonlinear mappings (of 1 onto 1, for instance) whose graph is closed although they are not continuous: f(x)= 1/x if x0, f(0)=0.
  17. If μ is a ositive measure, each f L(μ) defines a multiplication operator Mf on L2(μ) into L2(μ), such that Mf(g) =fg. Prove that Mf f. For which measures μ is it true that Mf = f? For which f L(μ) does Mf map L2(μ) onto L2(μ)?
  18. Suppose {Λn} is a sequence of bounded linear transformations from a normed linear space X to a Banach space Y, suppose Λn M< for all n, and suppose there is a dense set EX such that {Λnx} converges for each xE. Prove that {Λnx} converges for each xX.
  19. If sn is the nth partial sum of the Fourier series of a function fC(T), prove that sn/ logn0 uniformly, as n, for each f C(T). That is, prove that
    limn sn logn =0.
    On the other hand, if λn/ logn0, prove that there exists an fC(T) such that the sequence { sn(f;0)/ λn} is unbounded. Hint: Apply the reasoning of Exercise 18 and that of Sec. 511, with a better estimate of Dn1 than was used there.
  20. (a)  Does there exist a sequence of continuous positive functions fn on 1 such that {fn(x)} is unbounded if and only if x is rational?
    (b)   Replace "rational" by "irrational" in (a) and answer the resulting question.
    (c)   Replace "{fn(x)} is unbounded" by "fn(x) as n" and answer the resulting analogues of (a) and (b).
  21. Suppose E1 is measurable, and m(E)=0. Must there be a translate E+x of E that does not intersect E? Must there be a homeomorphism h of 1 onto 1 so that h(E) does not intersect E?
  22. Suppose fC(T) and fLipα for some α>0. (See Exercise 11.) Prove that the Fourier series of f converges to f(x), by completing the following outline: It is enough to consider the case x=0, f(0) =0. THe difference between the partial sums sn( fn;0) and the integrals
    1π -ππ f(t) sinnt t dt
    tends to 0 as n. The function f(t)/t is in L1(T). Apply the Riemann-Lebesgue lemma. More careful reasoning shows that the convergence is actually uniform on T.

Notes and References

These exercises are taken from [Ru, Chapt. 5] 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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