Let $(K, \tau_K)$ be a topological space and $(K, +,-, \times)$ be a field. $(K, \tau_K, +, -,\times )$ is called a topological field if $(a,b) \mapsto a+b$, $(a,b) \mapsto a \times b$ and $a\mapsto a^{-1}$ are continuous. Let $E$ be a vector space over a topological field $K$. By $E^*$ we denote the set of $K$-linear maps of $E$ into $K$. If $x\in E$ and $y^*\in E^*$ then denote $\langle x, y^*\rangle=y^*(x)$. By the $K$-linear topology of $E$ is meant the least topology of $E$ with respect to which each element of $E^*$ is contininous.
Assume that $\varphi:K\times K\to K$ by $(k, k')\mapsto kk'$ and $\psi:K\times E\to E$ by $(k, x)\mapsto kx$.
What can say about continuity of $\varphi$ and $\psi$?
Is it true that for evry $k\in K$, $\varphi_k:K\to K$ by $k'\mapsto kk'$ is an open map?