Suppose that $\forall j\in J: X_j $ is a locally convex space, with defining family of seminorms $(q_{jk})_{k \in K_j}$. Also let $X$ be a vector space and $T_j: X \to X_j$ a linear map $\forall j\in J$.

Why does the initial topology on X w.r.t. the family of linear maps $(T_j)_j$ coincide with the locally convex topology defined by the seminorms $\{q_{jk} \circ T_j | j \in J, k \in K_j \}$?

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    $\begingroup$ Because the $T_j$-preimage of a $q_{jk}$-ball is a $(q_{jk}\circ T_j)$-ball. $\endgroup$ – Daniel Fischer Sep 29 '14 at 21:31
  • $\begingroup$ Can you be a bit more precise, i.e. give a little bit more details please? $\endgroup$ – user179778 Sep 29 '14 at 21:35
  • $\begingroup$ You probably know that a vector space topology is completely determined by the family of neighbourhoods of $0$. A neighbourhood basis of $0$ in the initial topology is given by finite intersections of the $T_j$-preimages of neighbourhood bases of $0$ in $X_j$. For the neighbourhood basis in $X_j$, take the family of (finite intersections of, if the family $\{q_{jk} : k \in K_j\}$ is not filtering) $q_{jk}$-balls. $\endgroup$ – Daniel Fischer Sep 29 '14 at 21:40

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