This preview shows pages 1–2. Sign up to view the full content.
F2010. Homework # 9.
Section 4.2
7
a) The set of continuous real valued functions on
R
is not empty. It contains at
least the function constant equal to 0.
The sum of two continuous functions is continuous.
If we consider the function constant and equal to
c
, this function is continuous. If
f
is any continuous function from
R
to
R
, the product
cf
is continuous as product of
continuous functions.
The set of continuous real valued functions on
R
is not empty, closed for addition and
scalar multiplication, therefore it is a subspace.
b) Exactly as a) by replacing
continuous
by
diﬀerentiable
.
c) The set of real valued functions on the interval [0
,
1] such that
f
(1
/
2) = 0 is not
empty. It contains at least the function constant equal to 0.
If
f
and
g
are real valued functions on the interval [0
,
1] such that
f
(1
/
2) = 0 and
g
(1
/
2) = 0, their sum is a real valued functions on the interval [0
,
1], and (
f
+
g
)(1
/
2) =
f
(1
/
2) +
g
(1
/
2) = 0.
If
f
is a real valued functions on the interval [0
,
1] such that
f
(1
/
2) = 0 and
c
is a real
number,
cf
is a real valued functions on the interval [0
,
1], and (
cf
)(1
/
2) =
cf
(1
/
2) = 0.
The set of real valued functions
f
on the interval [0
,
1] such that
f
(1
/
2) = 0 is not
empty, it is closed for addition and scalar multiplication, therefore it is a subspace.
Remark.
The set of real valued functions on the interval [0
,
1] such that
f
(1
/
2) =
a, a
̸
= 0 is not a subspace. (0 does not belong to it, it is not closed for addition or scalar
multiplication.)
d) The set of real valued functions on the interval [0
,
1] such that
∫
1
0
f
(
x
)
dx
= 0 is
not empty. It contains at least the function constant equal to 0.
If
∫
1
0
f
(
x
)
dx
= 0 and
∫
1
0
g
(
x
)
dx
= 0, then
∫
1
0
(
f
+
g
)(
x
)
dx
=
∫
1
0
f
(
x
)
dx
+
∫
1
0
g
(
x
)
dx
= 0.
So closed for addition.
If
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview. Sign up
to
access the rest of the document.
 Spring '08
 HIETMANN

Click to edit the document details