If X is a normed linear space ,$x_{1},x_{2},…,x_{n}\in X$ are linear independent,$a_{1},a_{2},…,a_{n}\in F$ are arbitrary,then there exist $f\in X^{\ast } $ such that $f\left( x_{k}\right)= a_{k}$,$k=1,2,…,n$.
I have tried to find a $M\in F$ such that for aritrary $t_{1},t_{2},…,t_{n}\in F$,$\left| \sum _{k=1}^{n} t_{j}a_{j}\right|\leq M\left\|\sum _{k=1}^{n} t_{j}x_{j}\right\|$ but it seemed failed.
Is it necessary to show that $\overline {f}$ is bounded?$\overline {f}\in span\ \left\{ x_{1},x_{2},…,x_{n}\right\}$ $\overline {f}\left( x_{k}\right)= a_{k}$