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Let $X$ be a set and $\{(Y_i, \mathscr{T}_i)\}_{i\in I}$ be a family of topological spaces and $\{f_i\}_{i\in I}$ a family of mappings $$f_i:X\longrightarrow Y_i.$$ The initial topology on $X$ is the coarsest topology on $X$ making all the maps $f_i$ continuous. It is not hard to show that the set $$\mathscr{S}:=\{f^{-1}_i(U): U\in\mathscr{T}_i\}$$ is a sub-basis for this topology.

I want to apply this in the context of topological vector spaces.

The topology generated by a family of semi-norms $\mathscr{P}=\{p_i\}_{i\in I}$ on a $\mathbb R$-vector space is the initial topology on $X$ with respect to the family of mappings $\{p_i\}_{i\in I}$.

Can anyone help me showing this topology makes the vector space operations continuous?

Sketch (for the vector addition): We also know, from the general theory of initial topologies, that given a topological space $Z$ and a map $g:Z\longrightarrow X$ then $$g\ \textrm{is continuous}\Leftrightarrow g\circ f_i\ \textrm{is continuous}\ \forall i\in I.$$ Once this holds it suffices showing $p_i\circ g$ is continuous for all $i\in I$ where $g$ is the vector adition, this is where I'm stuck.

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If $p$ is a semi-norm, you have $0 \leq p(x-x' + y-y') \leq p(x-x')+p(y-y')$ for all $x,y$, showing that the initial topology $\mathscr{T}_p$ for $p$ make the addition continous. Now, the results follows, as if $(p_i)_{i \in I}$ is a family of semi-norms, the initial topology they generate is simply $\cap_{i\in I} \mathscr{T}_{p_i}$.

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