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 \(\mathbb{R}^{\mathbb{R}}\).