Note: 旧的wordpress博客弃用，于是将以前的笔记搬运回来。
Question:
A function
f
:
R
→
R
f : \mathbb{R} \rightarrow \mathbb{R}
f
:
R
→
R
is called periodic if there exists a positive number
p
p
p
such that
f
(
x
)
=
f
(
x
+
p
)
f(x) = f(x + p)
f
(
x
)
=
f
(
x
+
p
)
for all
x
∈
R
x \in \mathbb{R}
x
∈
R
. Is the set of periodic functions from
R
→
R
\mathbb{R} \rightarrow \mathbb{R}
R
→
R
a subspace of
R
R
\mathbb{R}^{\mathbb{R}}
R
R
? Explain.
UPD 2020/08/04: 下面的solution是错误的，感谢指正，正解可参考
https://math.stackexchange.com/questions/1764349/is-the-set-of-periodic-functions-from-mathbbr-to-mathbbr-a-subspace-of
或
https://linearalgebras.com/1c.html
Solution:
The answer to this question is YES. Now we will prove it.
Let
V
V
V
be the set of all periodic functions from
R
→
R
\mathbb{R} \rightarrow \mathbb{R}
R
→
R
.
Part 1:
Let
f
0
(
x
)
=
0
f_0(x) = 0
f
0
(
x
)
=
0
. Clearly
f
0
∈
V
f_0 \in V
f
0
∈
V
.
Part 2:
Take
f
,
g
∈
V
f, g \in V
f
,
g
∈
V
,
f
(
x
+
m
)
=
f
(
x
)
f(x + m) = f(x)
f
(
x
+
m
)
=
f
(
x
)
,
g
(
x
+
n
)
=
g
(
x
)
g(x + n) = g(x)
g
(
x
+
n
)
=
g
(
x
)
.
Clearly,
(
f
+
g
)
(
x
+
l
c
m
(
m
,
n
)
)
=
f
(
x
+
l
c
m
(
m
,
n
)
)
+
g
(
x
+
l
c
m
(
m
,
n
)
)
=
f
(
x
)
+
g
(
x
)
=
(
f
+
g
)
(
x
)
\begin{aligned}(f + g)(x + lcm(m, n)) = & f(x + lcm(m, n)) + g(x + lcm(m, n)) \\ = & f(x) + g(x) \\ = & (f + g)(x)\end{aligned}
(
f
+
g
)
(
x
+
l
c
m
(
m
,
n
)
)
=
=
=
f
(
x
+
l
c
m
(
m
,
n
)
)
+
g
(
x
+
l
c
m
(
m
,
n
)
)
f
(
x
)
+
g
(
x
)
(
f
+
g
)
(
x
)
.
Thus
f
+
g
∈
V
f + g \in V
f
+
g
∈
V
.
Part 3:
Take
f
∈
V
f \in V
f
∈
V
, and
a
∈
R
a \in R
a
∈
R
.
We have
(
a
f
)
(
x
+
p
)
=
a
⋅
f
(
x
+
p
)
=
a
⋅
f
(
x
)
=
(
a
f
)
(
x
)
(af)(x + p) = a \cdot f(x + p) = a \cdot f(x) = (af)(x)
(
a
f
)
(
x
+
p
)
=
a
⋅
f
(
x
+
p
)
=
a
⋅
f
(
x
)
=
(
a
f
)
(
x
)
.
Thus
a
f
∈
V
af \in V
a
f
∈
V
.
Therefor the set of all periodic functions from
R
→
R
\mathbb{R} \rightarrow \mathbb{R}
R
→
R
is a subspace of
R
R
\mathbb{R}^{\mathbb{R}}
R
R
.