I am reading the book by Eduard Zehnder, Lectures on dynamical Systems. I have a little question about one equation he gives.

He consider a $2n \times 2n$ matrix $J$ such that $J^2=-Id$. He then consider the space $C^{\infty}(S^1,\mathbb{R}^{2n})$, so the space od smooth loops I guess. On this space he defines $a(x,y)= \int_{0}^{1} \frac{1}{2}<-J \dot{x},y>dt$. In the book he gives the following Fourier representation $x(t)= \sum_{n \in \mathbb{Z}}e^{n 2 \pi t J }x_n$, where $x_n \in \mathbb{R}^{2n}$. If I understood correctly, the matrix $J$ mimic the imaginary unit $i$.

He then gives $\int_{0}^{1} <e^{n 2 \pi t J}x_n, e^{m 2 \pi t J}x_m> dt = \delta_{n,m}<x_n,x_m>$ which comes from the "orthogonal basis" property of the complex exponentials I think.

Now, He says that $2a(x,y) = 2 \pi \sum_{n >0} n<x_n,y_n> - 2 \pi \sum_{n <0} \mid n \mid <x_n,y_n>$.

I am quite unsure how he got this result. Here's what I have done. At one point I interchanged the integral sign and the sum and I am not able to justify this. Could you please tell me if I am wrong ? Or if it is indeed correct, how can I justify it.

Many thanks

$\begin{align*} 2a(x,y) & = \int_{0}^{1} <-J \dot{x},y> dt \\ & = \int_{0}^{1} <-J \cdot J \cdot 2 \pi \sum_{n \in \mathbb{Z}} e^{2 \pi n t J} x_n n , \sum_{m \in \mathbb{Z}} e^{2 \pi t J} y_m > dt \\ & = 2 \pi \int_{0}^{1} < Id \sum_{n \in \mathbb{Z}}n x_n,\sum_{m \in \mathbb{Z}} y_m >dt\\ & = 2 \pi \sum_{n,m \in \mathbb{Z}} \int_{0}^{1} < e^{2 \pi n t J} x_n n,e^{2 \pi t J} y_m > dt \\ & = 2 \pi \sum_{n \in \mathbb{Z}} n<x_n,y_n> \\ & = 2 \pi \sum_{n >0} n<x_n,y_n> - 2 \pi \sum_{n <0} \mid n \mid <x_n,y_n> \end{align*}$


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