Two equivalent statements for a sequence of $L^p$ functions This problem showed up on a previous qualifying exam and I have had issues trying to solve it. 

Let $(X,\cal{M},\mu)$ be a positive measure space. Let $p,q\in(1,\infty)$ satisfy $\frac{1}{p}+\frac{1}{q} = 1$. Let $(g_n)_{n\in \mathbb{N}}$ be a sequence in $L^p(\mu)$. Prove the following are equivalent:
(1) For any $f\in L^q(\mu)$, we have $$   \sum_{n=1}^\infty \left|\int_X
        g_n(x)f(x)\,d\mu\right|<\infty. $$
(2)   There exists $M\in[0,\infty)$ such that for any sequence
      $(\epsilon_k)_{k=1}^n$ in $\mathbb{C}$ with $|\epsilon_k|=1$,
      $k=1,\dots,n$, we have $$   \left\|\sum_{k=1}^n \epsilon_k g_k
    \right\|_p \leq M. $$

My thoughts: I'm to think that this $n$ in condition $(2)$ is arbitrary, meaning that we can take arbitrarily many unit-norm numbers and the condition in $(2)$ is satisfied.
Most of my progress is on the direction $(1) \implies (2)$. Condition $(1)$ screams uniform boundedness principle to me. Using the isomorphism $L^p \simeq (L^q)^*$, the statement in $(1)$ could be recast as
$$
  \sum_n |\Lambda_n(f)| < \infty, \quad \forall f \in L^q,
$$
where $\Lambda_n(f) = \int g_n f \,d\mu$ is the corresponding bounded linear functional to $g_n$. I then wish to consider the functionals
$$
  T_n: L^q \to \mathbb{C}, \quad T_n(f) = \sum_{k=1}^n \Lambda_k(f).
$$
Given that $(T_n(f))_{n\in \mathbb{N}}$ is a convergent sequence in $\mathbb{C}$ for each $f \in L^q$, this sequence is bounded in $\mathbb{C}$. Then, uniform boundedness must imply that
$$
  \sup_n ||T_n||_{(L^q)^*}<\infty.
$$
In other words, there exists $M<\infty$ such that
$$
  \left\|\sum_{k=1}^n \Lambda_k \right\|_{(L^q)^*} \leq M.
$$
The isomorphism should imply that
$$
  \left\|\sum_{k=1}^n g_k\right\|_p \leq M
$$
as well. But I fail to see how to get the above bound when I put in $\epsilon_k$ next to each $g_k$. Perhaps some form of Cauchy-Schwarz? 
I have ideas for the other direction, but nothing's panned out. I note that for $f\in L^q$, the Holder inequality gives
$$
 \left |\int f g_n \,d\mu \right|\leq \|f\|_q \|g_n\|_p,
$$
so 
$$
  \sum_n \left|\int_X f g_n \,d\mu\right| \leq \|f\|_q \sum_n \|g_n\|_q.
$$
If I can use the statement in $(2)$ to prove the summability of the series on the right, I would be done. 
Can someone shed some light on how to do finish off either of these directions?
 A: Assume (1) holds. As per Daniel Fischer's comment, consider the family
$$
  \mathcal{F} = \{\sum_{k=1}^n \epsilon_k g_k: n\in \mathbb{N}, (\epsilon_k) \in (S^1)^{\mathbb{N}}\}.
$$
For $h\in \mathcal{F}$, let $\Lambda_h(f) = \int h f\,d\mu $ be the corresponding bounded linear functional in $(L^p)^*$. Then
$$
  |\Lambda_h(f) | \leq \sum_{k=1}^n \left|\epsilon_k \int f g_k\right| \leq \sum_{k=1}^\infty \left|\int f g_n \,d\mu \right| <\infty \qquad \forall h\in \mathcal{F}
$$
by the assumption in $(1)$. So, by Banach-Steinhaus,, we have
$$
 M:= \sup_{h\in \mathcal{F}} \|\Lambda_h\| < \infty.
$$
We have that $\|\Lambda_h \| = \| h\|_p$, so
$$
  \left\|\sum_{k=1}^n \epsilon_k g_k\right\|_p \leq M
$$
for all suitable choices of $(\epsilon_k)$.
Conversely, assuming $(2)$, let $f\in L^q$. As per A.G.'s comment, choose $\epsilon_k$ such that $\epsilon_k \int f g_k \,d\mu = \left|\int fg_k\,d\mu \right|$. Then, statement $(2)$ implies that
$$
  \sum_{k=1}^n \left|\int f g_k\,d\mu  \right| = \sum_{k=1}^n \epsilon_k \int fg_k\,d\mu = \int f \left(\sum_{k=1}^n \epsilon_k g_k \right)\,d\mu \leq \left\|\sum_{k=1}^n \epsilon_k g_k\right\|_p \|f\|_q \leq M \|f\|_p,
$$
By Holder's inequality. Thus, the partial sums of the series of positive terms are bounded, and hence the series converges.
