I'm working on this exercise (not homework) and I would gladly welcome some hints for how to solve it!
Exercise: Let $\{x_1,\dots,x_n\}$ be a set of linearly independent elements of a normed vector space $X$. Let $c_1,\dots,c_n \in \mathbb{C}$. Show that there exists $f\in X^\ast$ such that $f(x_i)=c_i$.
My idea:
I consider $M = span\{x_1,...,x_n\}$, which is a subspace of $X$. Any $x\in M$ can be written $x=\sum_1^n \lambda_k x_k$, for some $\lambda_1,...,\lambda_n \in \mathbb{C}$. Define $f:M \rightarrow \mathbb{C}$ by $f(x_i)=c_i$ for $i=1,...,n$. Then $f(x) = \sum_1^n \lambda_k f(x_k) = \sum_1^n \lambda_k c_k.$
If I can find a semi-norm $p:X \rightarrow \mathbb{R}$ such that $|f(x)| \leq p(x)$ for any $x \in M$, then by Hahn-Banach Theorem we would be done.
Thanks in advance!