Fermi
·
Concept Library
·
Courses
·
Books
·
Problems
·
Simulations
·
Math Wiki
·
High School
Fermi
›
Problems
Basic Analysis I — Real Analysis
· Problem 7.1.7
Metric Spaces — Jiri Lebl
Mathematics — Real Analysis · metric-spaces
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) \coloneq \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
metric-spaces
Browse the problem library