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Lebl-Basic-Analysis-I
· Problem 7.1.7
Metric Spaces
Mathematics · metric-spaces-definition
Problem
Let
X
X
X
be the set of continuous functions on
[
0
,
1
]
[0, 1]
[
0
,
1
]
. Let
φ
:
[
0
,
1
]
→
(
0
,
∞
)
\varphi \colon [0, 1] \to (0, \infty)
φ
:
[
0
,
1
]
→
(
0
,
∞
)
be continuous. Define
d
(
f
,
g
)
≔
∫
0
1
∣
f
(
x
)
−
g
(
x
)
∣
φ
(
x
)
d
x
.
d(f, g) \colonequals \int_0^1 \bigl|f(x) - g(x)\bigr| \varphi(x)\, dx.
d
(
f
,
g
)
:
=
∫
0
1
f
(
x
)
−
g
(
x
)
φ
(
x
)
d
x
.
Show that
(
X
,
d
)
(X, d)
(
X
,
d
)
is a metric space.
Related theory
Continuous Maps Between Metric Spaces
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