Definition 8.5.4 (Product Topology). Let $\seqi{E}$ be TVSs over $K \in \RC$ and $E = \prod_{i \in I}E_{i}$ be their product as a vector space, and $\fU$ be the initial uniformity generated by the projection maps, then
$E$ equipped with the topology induced by $\fU$ is a topological vector space.
For any TVS $F$ over $K$ and $\seqi{T}$ where $T_{i} \in L(F; E_{i})$ for each $i \in I$, there exists a unique $U \in L(F; E)$ such that the following diagram commutes
\[\xymatrix{ F \ar@{->}[rd]^{T_i} \ar@{->}[d]_{T} & \\ \prod_{i \in I}E_i \ar@{->}[r]_{\pi_i} & E_i }\]
The uniformity $\fU$ and its induced topology are the product uniformity/topology, and $E$ equipped with $\fU$ is the product TVS of $\seqi{E}$.