## R到R的周期函数构成的集合是R^R的子空间吗？

Question:

A function $$f : \mathbb{R} \rightarrow \mathbb{R}$$ is called periodic if there exists a positive number $$p$$ such that $$f(x) = f(x + p)$$ for all $$x \in \mathbb{R}$$. Is the set of periodic functions from $$\mathbb{R} \rightarrow \mathbb{R}$$ a subspace of $$\mathbb{R}^{\mathbb{R}}$$? Explain.

Solution:

The answer to this question is YES. Now we will prove it.

Let $$V$$ be the set of all periodic functions from $$\mathbb{R} \rightarrow \mathbb{R}$$.

Part 1:

Let $$f_0(x) = 0$$. Clearly $$f_0 \in V$$.

Part 2:

Take $$f, g \in V$$, $$f(x + m) = f(x)$$, $$g(x + n) = g(x)$$.

Clearly, \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}.

Thus $$f + g \in V$$.

Part 3:

Take $$f \in V$$, and $$a \in R$$.

We have $$(af)(x + p) = a \cdot f(x + p) = a \cdot f(x) = (af)(x)$$.

Thus $$af \in V$$.

Therefor the set of all periodic functions from $$\mathbb{R} \rightarrow \mathbb{R}$$ is a subspace of $$\mathbb{R}^{\mathbb{R}}$$.