\begin{equation} P = \{f \in\ C ( \mathbb{R}, \mathbb{R}) \mid f(x+2 \pi ) = f(x)\} \end{equation} be the set of $2\pi$-periodic function.
1) Show that $P$ is a subspace of $C( \mathbb{R}, \mathbb{R})$
2) Let $\|F\| = \sup_{x\in[0,2\pi]} |f(x)| $ for all $ f\in P$. Show that $ (P, \| \cdot \|)$ is a normed space.
3) Let $d(f,g) = \| f - g \| $ for all $f,g\in P$. Show that $d$ is a metric and $(P,d)$ is a complete metric space.
I have an exam soon and this was one of the practice questions.
For 1) part: I know that for $P$ to be subspace of $C( \mathbb{R}, \mathbb{R})$, I have to show that it is closed under vector addition and scalar multiplication, right? But how?
For 2) part: I can prove it using the axioms for a normed space.
For 3) part: I proved that $d$ is a metric by using axioms, but then how do I prove that $(P,d)$ is a complete metric space? I know that a metric space is called complete if every Cauchy sequence in that space is convergent, but how do I prove it for this part?