linear independence in a dual pair

Question

The following is an excerpt from the Topological Vector Spaces by Schaefer:

I don't see how the underscored sentence work. Suppose $$ \langle x,y_n\rangle=0 $$ for all $x\in F_n$. Why this implies $$ y_n\in\hbox{span}\{y_1,y_2,\cdots,y_n\}? $$ I guess it might have something to do with the direct sum $F=F_n+M_n$. But I don't see the connection.

Answer 1

Consider $y' = y_n - \langle \overline{x}_1,y_n\rangle y_1 - \langle \overline{x}_2,y_n\rangle y_2 - \cdots - \langle \overline{x}_{n-1},y_n\rangle y_{n-1}$.

1. Can you prove that $y'$ vanishes on all $\overline{x}_i$ and therefore on $M_n$?

2. Can you prove that $y'$ vanishes on all $x \in F_n$?

Now use the direct sum and conclude what $y'$ must be, and what that implies.