# Using convexity and separation to prove bounds on norm bound functionals

I'm quoting here a homework problem with two clauses. I've already managed to find a solution for the first clause, and have problems generalizing it for the second clause, I'll go into details after the problem specification.

a. Let $X$ be a real linear space with the norms $\{\|\cdot \| \}_{i=1}^n$ and let $f$ be a linear functional on $X$ such that $\forall x\in X:f(x)\le \max_{1\le i\le n}\|x\|_i$.

Prove the existence of non negative real scalars $\{\lambda_i\}_{i=1}^n$ s.t. $\sum\lambda_i=1$ and $\forall x\in X:f(x)\le \sum_{1\le i\le n}\lambda_i\|x\|_i$

b. Let $X$ be a real linear space with the norms $\{\|\cdot \| \}_{i=1}^\infty$. Let us assume that for any $x\in X$ the limit $|\|x|\|=\lim_{i\to\infty}\|x\|_i$ exists and is finite.

Let $\varphi$ be a linear functional on $X$ such that $\forall x\in X: \varphi(x)\le\sup\|x\|_i$.

Prove that there are non-negative scalars $\{\lambda_i\}_{i=0}^{\infty}$ s.t. $\sum\lambda_i=1$ and $\forall x\in X:\varphi(x)\le\lambda_0|\|x|\|+\sum_{i=1}^\infty \lambda_i\|x\|_i$.

To solve the first clause I defined the set $D=\{(...,\|x\|_i-f(x),...|x\in X\}$, and I used the fact that each element has at least one non-negative coordinate to separate it from the set of vectors with all negative coordinates in $\mathbb{R}^n$. Using that I was able to use a separation theorem, which produced a separating functional. And since this space is finite, I could use Riesz' representation to get a vector which separates them through inner product and from there construct the scalars.

My TA told me this approach should be generalized to handle the second clause, where instead of $\mathbb{R}^n$ I use $c$ (the space of converging sequences). However, I run into several difficulties. The first one, is proving that in this infinite case, the convex hull of the set still only contains point with at least one non-negative coordinate. To prove it for the finite case, I just had to show it holds for any finite convex combination, and it followed immediately from Caratheodory's convex hull theorem. The second one is that, since the space is not finite this time, I'm not at liberty to use Riesz' representation theorem on the separating functional.

I've already proven before that $c^\star$ is isomorphic to $\ell^1$ via $Tx(y)=x_1\lim y_n +\sum_{n=2}^\infty x_n y_{n-1}$, and I suspect that might be useful, I'm just not sure how.