continuous functional with norm less than or equal to 1

Question

Let $\left\{ x_n\right\}_{n=1}^\infty$ be a sequence of vectors in a normed space $X$, such that $\sup_{n\ge 1} \Vert x_n\Vert\le 1$ and let $\left\{ \varepsilon_n\right\}_{n=1}^\infty$ be a sequence of non-negative real numbers, such that $\sum_{n=1}^{\infty}\varepsilon_n=1$. Show, that there exists $\varphi_0\in X'$ satisfying $\Vert \varphi_0\Vert\le 1$ and

$\sum_{n=1}^{\infty}\varepsilon_n|\varphi(x_n)|\le\sum_{n=1}^{\infty}\varepsilon_n|\varphi_0(x_n)|$ , $ \ \ \ \ $ $\varphi\in X',\Vert \varphi\Vert=1$.

And the second part of this excercise:

Show that for any sequence $\left\{\zeta_n\right\}_{n=1}^{\infty}\subset \mathbb{K}$ satisfying $|\zeta_n|\le \varepsilon_n$ for all $n\ge 1$ we have

$\Vert\sum_{n=1}^{k}\zeta_n x_n\Vert\le\sum_{n=1}^{\infty}\varepsilon_n|\varphi_0(x_n)|$, for $k=1,2,...$

I have no idea even how to start this problem, can you give me any hint?

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Assuming the first part of excercise I think the second one is following:

$\Vert \sum_{n=1}^k \zeta_n x_n\Vert\le\sum_{n=1}^k\Vert\zeta_n x_n\Vert=\sum_{n=1}^k|\zeta_n|\Vert x_n\Vert\le\sum_{n=1}^k\varepsilon_n\Vert x_n\Vert\le\sum_{n=1}^k\varepsilon_n |\varphi(x_n)|\le\sum_{n=1}^\infty\varepsilon_n |\varphi(x_n)|\le\sum_{n=1}^\infty\varepsilon_n |\varphi_0(x_n)|$

Is this ok? (2nd part)

Answer 1

The first question is a consequence of Alaoglu. The function defined on the closed unit$B^*$ ball by $H(\phi)=\sum\epsilon_n|\phi(x_n)\|$ is continue, since $B^*$ is weakly compact $x_0$ exists.

https://en.wikipedia.org/wiki/Banach%E2%80%93Alaoglu_theorem#The_theorem