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Lebl-Basic-Analysis-I
· Problem 7.4.11
Metric Spaces
Mathematics · compactness
Problem
(Challenging) Let
C
(
[
0
,
1
]
,
R
)
C\bigl([0, 1], \mathbb{R}\bigr)
C
(
[
0
,
1
]
,
R
)
be the metric space of Example 7.1.8. Let
K
K
K
be the set of
f
∈
C
(
[
0
,
1
]
,
R
)
f \in C\bigl([0, 1], \mathbb{R}\bigr)
f
∈
C
(
[
0
,
1
]
,
R
)
such that
f
f
f
is equal to a quadratic polynomial, i.e.
f
(
x
)
=
a
+
b
x
+
c
x
2
f(x) = a + bx + cx^2
f
(
x
)
=
a
+
b
x
+
c
x
2
, and such that
∣
f
(
x
)
∣
≤
1
|f(x)| \leq 1
∣
f
(
x
)
∣
≤
1
for all
x
∈
[
0
,
1
]
x \in [0, 1]
x
∈
[
0
,
1
]
, that is
f
∈
C
(
0
,
1
)
f \in C(0, 1)
f
∈
C
(
0
,
1
)
. Show that
K
K
K
is compact.
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