Transform Fourier of paths in symplectic vector spaces

Question

In the proof of Lemma 4.4.4 in "J-holomorphic Curves and Symplectic Topology" by McDuff and Salamon, They state the two following claims:

Let $(V,\omega)$ be a symplectic vector space, and $J$ be an $\omega$-compatible a.c.s, and $L \subset V$ be a Lagrangian subspace, Then:

1. Every loop $\gamma:S^1 \rightarrow V$ can be written as a Fourier series:

$\ \ \ \begin{align}\gamma(\theta) =\sum_{k=−\infty}^{\infty}e^{k\theta J}v_k\end{align}$

2. Every path $\gamma:[0,\pi] \rightarrow V$ with endpoints in L can be written as a Fourier series with coefficients $u_k \in L$:

$\ \ \ \begin{align}\gamma(\theta) =\sum_{k=−\infty}^{\infty}e^{k\theta J}u_k\end{align}$

My two question regarding these claims are (and the first one bugs me more!):

1. In the second claim, why are the coefficients in $L$?

2. Can someone link any source on such Fourier series (in vector spaces, using a.c.s instead of $i$)?