Continuity of $x+y$ and $xy$ in $\mathbb{R}^{\infty}$ How can I show (or where can I find) that in $\mathbb{R}^\infty$: $f(\textbf{x},\textbf{y})=\textbf{x}+\textbf{y}$, $g(\textbf{x}, k)=k\cdot \textbf{x}$ are continuous functions?
($g$ is from $\mathbb{R}^\infty \times \mathbb{R}$ into $\mathbb{R}^\infty$).
 A: As the sets of the form $\prod_{j=1}^NO_j\times \Bbb R\times \cdots$, $N\in\Bbb N$, $O_j$ open subset of $\Bbb R$, it's enough to see that $f^{-1}$ and $g^{-1}$ of such sets is open in the product topology of the respective source sets. 
So fix $N\in\Bbb N$, and $O_j, 1\leqslant j\leqslant N$ open subsets of the real line. Let $O:=\prod_{j=1}^NO_j\times \Bbb R\times \cdots$. We have that 
$$\small f^{-1}(O)=\{(\mathbf x,\mathbf y)\in\Bbb R^\infty\times\Bbb R^\infty, \forall j\in [N], x_j+y_j\in O_j\}=\bigcap_{j=1}^N\{(\mathbf x,\mathbf y)\in\Bbb R^\infty\times\Bbb R^\infty, x_j+y_j\in O_j\},$$
so it's enough to see that $S:=\{(\mathbf x,\mathbf y)\in\Bbb R^\infty\times\Bbb R^\infty, x_1+y_1\in O_1\}$ is open. For $a,b\in\Bbb R^2$, denote $\phi(a,b):=a+b$. It defines a continuous map on $\Bbb R^2$. So $S=\{(\mathbf x,\mathbf y)\in\Bbb R^\infty\times\Bbb R^\infty, (x_1,y_1)\in \phi^{-1}(O_j)\}$. By definition of the product topology, the map $\pi_j\colon \Bbb R^\infty\times\Bbb R^\infty\to \Bbb R^2$ given by $\pi_j(\mathbb x,\mathbb y):=(x_j,y_j)$ is continuous, proving that so is $f$. 
It's the same idea for $g$. 
