Compatibility of product topology with the vector space structure I've been struggling with the following rudimentary question and I'd appreciate any help :  
Given topological $K$-vector spaces $(E_i)_{i \in I}$, let $E$ be their product and $T$ the product topology on $E$. How exactly do we prove that $T$ is compatible with the vector space structure of $E$, that is how do we prove that the mappings $f:E\times E\rightarrow E,\ (x,y)\mapsto x+y$ and $g:K\times E\rightarrow E,\ (\lambda,x)\mapsto \lambda x\ $ \ are both continuous ?
Thank you in advance for your answers !
 A: Firstly, we can identify (via the obvious homeomorphism) $E \times E \cong \prod_{i=1}^n E_i \times E_i$. If $\oplus:E \times E \to E$ is addition in $E$ and $\oplus_i: E_i \times E_i \to E_i$ is addition in $E_i$ then under the above identification we have $\oplus = \prod_{i=1}^n \oplus_i$.
Since $\oplus_i$ is continuous for each $i$, it suffices to show that a product of continuous maps is continuous. This is a standard topological result. Indeed, if $f_i: X_i \to Y_i$ are continuous maps of topological spaces then for a basic open set $U_1 \times \dots \times U_k \subseteq \prod_{i=1}^k Y_i$ we have $$(f_1 \times \dots \times f_k)^{-1}(U_1 \times \dots \times U_k) = \prod_{i=1}^k f_i^{-1}(U_i)$$
which is a basic open set in $\prod_{i=1}^k X_i$ and hence $f_1 \times \dots \times f_k$ is continuous.
The same idea will also give the second result, by identifying $\prod_{i=1}^n K \times E_i \cong K^n \times E$ and noting that $\otimes: K \times E \to E$ is then a restriction of the product map $\prod_{i=1}^n \otimes_i$ (up to a composition with the obvious embedding $K \hookrightarrow K^n$ given by $\lambda \mapsto (\lambda, \dots, \lambda)$).
