Let $(V,\|\cdot\|)$ be a [normed space](/page/Normed%20Space) over a scalar field $\mathbb{F}\in\{\mathbb{R},\mathbb{C}\}$. Equip $V$ with the norm topology, equip $V\times V$ with the [product topology](/page/Product%20Topology), and equip $\mathbb{F}$ with the topology induced by the absolute value $|\cdot|$. Define the addition map $A:V\times V\to V$ by $A(x,y)=x+y$, and define the scalar multiplication map $M:\mathbb{F}\times V\to V$ by $M(\lambda,x)=\lambda x$. Then both $A$ and $M$ are continuous.