Dual space of $R^n$ with $\max$ norm

Question

So I think it's supposed to be the $(R^n,||.||_1)$, but I can't get one of the inequalities: So if $\lambda_k = f(e_k)$, for $f$ in the dual, $x = (x_1,...,x_n) \in R^n$ and $\{e_k\}$ basis, then: $$|f(x)| \leq \sum|x_k||\lambda_k| \leq \max|x_k|\sum|\lambda_k|$$ Thus $\||f||\leq \sum |\lambda_k|$ But I don't see how to get the reverse inequlity...

Answer 1

Consider a specific vector $x$ with $x_i=-1$ if $\lambda_i<0$ else $x_i=+1$. Then $\|x\|_\max=1$ and $$f(x) =\sum_ix_if(e_i) =\sum_i|\lambda_i|$$