ANALYSIS I: PROBLEM SET # 7
DUE FRIDAY, 22 APRIL
Notation. For any topological space X , denote by B(X ) the set of bounded, continuous functions f : X
this set, we have the uniform metric given by
R. On
d ( f , g ) := sup |g (x) − f (x)|.
x∈X
Recall that we have shown that this is a complete metric space, and in the topology induced by this metric, a
sequence ( fn )n≥0 of elements of B(X ) converges to an element f ∈ B(X ) if and only if ( fn )n≥0 converges uniformly
to f .
Exercise 52. Is every function f ∈ B(R m ) uniformly continuous?
Definition. Suppose X a topological space, and suppose E ⊂ X a subspace. Then the support of a function f : E
is the closure in X of the subset
{x ∈ E | f (x) 6= 0} ⊂ X .
R
Notation. Suppose X a topological space, and suppose E ⊂ X a subspace. We denote by supp( f ) the support of a
function f : E
X.
Suppose E1 , E2 ⊂ X two subspaces, and suppose f1 : E1
R and f2 : E2
R two functions. Then we write
f1 ≤ f2 (or f2 ≥ f1 ) if and only if supp( f1 ) ⊂ supp( f2 ), and for any point x ∈ supp( f2 ), one has f1 (x) ≤ f2 (x).
R whose support is compact.
Denote by C00 (X ) the set of continuous functions f : X
Exercise 53. Show that C00 (R m ) is a subspace of B(R m ), and describe its closure.
Exercise 54. Suppose S : C00 (R)
R a function with the following three properties.
(1) The function S is linear; that is, for any f , g ∈ C00 (R) and any γ ∈ R, one has
S( f + g ) = S( f ) + S(g ) and
S(γ f ) = γ S( f ).
C00 (R),
if f ≥ 0, then S( f ) ≥ 0, and if both f ≥ 0 and
(2) The function S is positive definite; that is, for any f ∈
S( f ) = 0, then f = 0.
(3) The function S is translation invariant; that is, for any f ∈ C00 (R) and any t ∈ R, one has S( f ) = S( t f ), where
t
f ∈ C00 (R) is defined by
t
f (x) = f (t + x).
Prove that there exists a positive real quantity α such that for any f ∈ C00 (R), one has
Zb
S( f ) = α
f,
a
where [a, b ] is a closed interval containing supp( f ).
Notation. A consequence of the previous exercise is that there is precisely one linear, positive definite, translation
invariant function I : C00 (R)
R with the property that I (h) = 1, where

if x ∈
/ [−1, 1];
0
h(x) = 1 + x if x ∈ [−1, 0];

1 − x if x ∈ [0, 1].
1
2
DUE FRIDAY, 22 APRIL
Exercise 55. Show that, for any Riemann integrable function g : [a, b ]
one has
Z
R such that g (x) ≥ 0 for every x ∈ [a, b ]
b
a
g = sup{I ( f ) | f ∈ C00 (R), f ≤ g }.
Exercise 56. Is I continuous?
Exercise 57. Suppose f , g ∈ C00 (R). Show that, for any u ∈ R, the function c f , g ,u : R
R defined by
c f ,g ,u (v) := f (v)g (u − v)
is an element of
C00 (R).
Show that f ∗ g
∈ C00 (R)
Now the convolution of f and g is defined as
( f ∗ g )(u) := I (c f , g ,u ).
as well. Deduce that the convolution product defines a map
C00 (R) × C00 (R)
C00 (R).
Exercise 58. Show that the convolution is associative and commutative; that is, for any f , g , h ∈ C00 (R),
f ∗ (g ∗ h) = ( f ∗ g ) ∗ h
and
f ∗g=g∗f.
Exercise 59. Prove that for any f , g ∈ C00 (R), one has
I ( f ∗ g ) = I ( f )I (g ).
Notation. For any nonnegative integer k, set
C0k (R) := C k (R) ∩ C00 (R).
Similarly, set
C0∞ (R) := C ∞ (R) ∩ C00 (R).
Exercise 60. Show that if f ∈ C01 (R) and g ∈ C00 (R), then f ∗ g ∈ C01 (R), and
( f ∗ g )0 = f 0 ∗ g .
Deduce that the convolution product restricts to a map
C0k (R) × C0` (R)
C0k+` (R).
Exercise 61. Use Exercise 51 to construct a function ϕ ∈ C0∞ (R) such that the following conditions hold.
(1) One has ϕ ≥ 0.
(2) One has I (ϕ) = 1.
(3) If ϕn is defined by
ϕn (x) := nϕ (n x) ,
then supp ϕn ⊂ B(0, 1/n).
For any f ∈ C00 (R), show that ϕn ∗ f → f uniformly as n → ∞. Deduce that, for any real number m, the subspace
of C0∞ (R) consisting of those f ∈ C0∞ (R) such that I ( f ) = m is dense in the subspace of C00 (R) consisting of those
f ∈ C00 (R) such that I ( f ) = m, and thus C0∞ (R) is dense in C00 (R).