Exercises

EXERCISES to Chapter I: 1. Prove that, if f is the characteristic function of any of bounded intervals (a, b), [a, b] or ( a , b], then it is integrable. 2. Prove that, if f is integrable, then also the function g such that g(x) = f ( - x ) is integrable. to Chapter 11: 1. Check that the following sets are linear spaces:

a) the set of the real numbers, b) the set of the complex numbers, c) the set of all continuous functions on the interval [0, 13, d) the set of all sequences of real numbers, tending to 0. 2. Check that, if by “addition” we mean the ordinary multiplication, by “multiplication by A ” the ordinary Ath power, then the set of all positive numbers is a linear space. 3. Show that the space of all polynomials with real coefficients is a linear normed space, if we take for the norm of the polynomial p the expression

)PI

= max Ip(x)l. Show that this space is not a Banach space, ro, 11

(Hint: show that it is not complete.) 4. Check that the set of all continuous functions on [0, 13 is a Banach space, if we take the norm If1 =max If(x)l. [O,

11

5 . Prove that, if a and b are elements of a Banach space, then either

la+-bl>lal or la-bI31al.

to Chapter 111:

1. The space of the complex numbers can be considered as a particular Banach space. Show that a complex valued function is Bochner integrable, if and only if its real part and its imaginary part are Lebesgue integrable. 2. Let L denote the space of the Lebesgue integrable functions. In particular, each brick function belongs to L. Let xx denote the brick function which is the characteristic function of the interval [ x , x + 1).

228

Exercises

Show that the function if XE[O, l), if x&[O, 1)

(whose values are in the Banach space L ) is Bochner integrable. What is its integral? to Chapter IV:

1. Prove that norm convergence satisfies Urysohn's condition L" to Chapter V: 1. Prove the following formulae, on reducing them to equivalent formulae but with ordinary algebraic operations: AUA=A, (A\B) n B = 0, (A nB)u (A\B) = A. 2. Prove that i.m., for each real A ; a) If fn+f i.m., then Af,,+Af b) If fn+f i.m., and gn+g i.m., then f n + g n + f + g i.m.; i.m. c) If fn+f i.m., then If,,l+lfl

to Chapter VI: 1. Prove that f.n (-f) = f. 2. Prove that if a sequence of measurable functions fn converges to f almost everywhere, then it converges to f locally in measure.

3. Prove that if f 2 0 ,then

J f s J f sJ f +

z,nzz

z,uz2

z,

If.

zz

4. Give an example of two functions f and g, f being continuous and g measurable, such that the superposition h(x) = g ( f ( x ) ) is not measurable. 5 . Show that a real valued function f is measurable, if and only if, for any real number a, the set of points where f < a is measurable. 1 6. Prove that if f is measurable and f Z 0 everywhere, then - is measurable. If1

to Chapter VII: 1. Show that a vector valued function f ( x ) is continuous at x, if and implies f(x,,)+f(x). In the proof, axiom of choice is only if x,+x needed. Where? 2. Show that if a set X of real numbers is measurable, then the set of points x such that x3 E X is also measurable.

Exercises

229

3. Show that if the outer cover of a set X is equivalent to its inner cover, then X is measurable. to Chapter VIII:

1. Let f(x)=- 1 if x =-,P where p and q are relatively prime positive

4 4 integers, and let f(x)=O if x is irrational. Prove that f is Riemann integrable over the interval (0, 1). 2. Prove that conditions C and C‘ in section 6 are equivalent.

to Chapter X: 1. Prove that if f is integrable over X and g is integrable over Y, then the product f ( x ) g ( y ) is integrable over X XY. 2. Prove that if the function f(t, u) of two variables t and u is integrable over the triangle A: 0 s t s u s T, then T

T

t

T

j d t j f ( t , u) du = j d u j f ( t , u) dt 0

u

0

u

(this formula is sometimes called the Dirichlet formula).

“t

to Chapter XI: 1. Prove that, if f(x, y ) is a function of two real variables x and y , continuous on a closed and bounded rectangle I: ( a s x s b, c S y C d ) , then b

a

d

d

c

c

b

n

(Hint: use the Fubini theorem). 2. Prove that if f is Bochner integrable over R’,then the functions

I

F(x)=

-m

are continuous.

I

m

m

sin x t . f ( r ) dt and

G(x) =

-m

cos x t f ( t ) dt

230

Exercises

to Chapter XII: 1. Prove that if f and g are functions of a real variable, integrable in (0, T), then the convolution

i

h ( t ) = f(t-u)g(u) du 0

T

is also integrable in (0, T). (Hint: consider the integral j h ( t ) dt and 0 apply the Tonelli theorem). 2. Prove that, if x E Rq and An denotes the set of the x such that 2-"-'< 1x1 < 2-", then

Deduce hence that the function lxlp is locally integrable in Rq, iff p >-q. 3. Show that an antiaureole of a square is another square and an aureole of a square never is a square. 4. Show that the function F ( x ) = x z sin with F(0) = 0 has its ordinary X

derivative 0 at x = 0, but the full derivative does not exist at x = 0. 1 5 . Show that the function F(x, y) = (xz+ y') sin -has a Stolz's x2+ y z differential at the origin, but is not fully derivable there. 6 . Interpret inequality (Jl) in the following one-dimensional case: X = [ - 1, 11, q ( x ) = x3- x. Show that the left side of the inequality is then 2 8 - and its right side is -. J3 3J3 to Chapter XIII: 1. Prove that the function 1x1 has, in R', its local derivative equal to sgn x. 1 for x s 0 , 2. Prove that the Heaviside function H ( x ) = has no 0 for x
the interval (0,l). Let f ( x ) =

m

1

n=l

2-"H(x-an). Prove that f is non-

decreasing in the interval (0, 1) and has no local derivative in any subinterval of (0,l).

Exercises

to Chapter XIV: 1. Prove that if f(x) = e-(x2’u2),g(x) = e-(x2’p2),then f

*

=

JW

e-x2/(e2+P2)

2. If H is the Heaviside function, then Xn-l

(H* .;. * H ) ( x )=n (n- l)!H ( x ) .

231