Proposition 7

The canonical injection $j : E \to \prod_{i \in I} E_i$ is continuous when $E$ carries the direct sum topology of the $E_i$ and $\prod_{i \in I} E_i$ carries the product topology. When $I$ is finite, this mapping is an isomorphism of topological vector spaces.

The first assertion follows from the fact that the canonical injections $E_\kappa \to \prod_{\iota \in I} E_\iota$ are continuous for each $\kappa \in I$. If $I$ is finite then $j$ is the identity mapping, and it is sufficient to show that the product topology $\mathscr{T}'$ is finer than the direct sum topology $\mathscr{T}$. Now, let $V$ be a convex neighbourhood of 0 for $\mathscr{T}$; each set $V \cap E_\iota$ is a convex neighbourhood of 0 in $E_\iota$; if $n$ is the number of elements of $I$, then the set $V$ contains the set $\frac{1}{n} \sum_n (V \cap E_\iota)$, which is a neighbourhood of 0 for $\mathscr{T}'$, and the proposition is proved.

When $I$ is infinite, if, for each finite subset $J$ of $I$, we write $E_J$ for the space $\prod_{\iota \in J} E_\iota$, with the product topology, then $E$ is the inductive limit of the $E_J$ (identified as subspaces of $E$).