1) Define : $\langle z\rangle := (1+|z|^2 ) ^\frac{1}{2} $ for all $z \in \mathbb{C} $. Prove : $\langle x+y\rangle \leq 2\langle x\rangle\langle y\rangle $ for all $x,y \in \mathbb{R} ^N$ .
2) Define: $ S^\beta := \left\{ f: \mathbb{R} \to \mathbb{C} : f \in C^\infty(\mathbb{R}), \ \left|f^{(n)}(x)\right|:= \left|\frac{d^n f}{dx^n}\right| \leq c_n \langle x\rangle^{\beta -n} \right\} $ for some $c_n<\infty$, all $x \in \mathbb{R} $ and all $ 0\leq n \in \mathbb{Z}$. Then, define: $ \mathbb{A} := \cup_{\beta<0 } S^\beta $. Prove $\mathbb{A}$ is an algebra under pointwise multiplication.
I'm pretty much clueless regarding (1). As for 2 - I can't understand which notion of an "algebra" they mean... If it's an algebra like in "sigma-algebra", then why do we need the pointwise multiplication? can you help me prove this?
Thanks