Reference for the sobolev space $H^{1/2}([0,1];\mathbb{R}^{n})$

Question

I need a reference for the Sobolev space $H^{1/2}([0,1]; \mathbb{R}^{n})$. The article that I am reading says that this space consists of all $L^2$ curves $\gamma: [0,1] → R^n$ such that the coefficients $(x_k)_{k\in Z}$ of the Fourier decomposition \begin{equation} \gamma (t) = \displaystyle\sum_{k\in \mathbb{Z}}e^{-2\pi kJt}x_k,\;\; x_k \in \mathbb{R}^n \end{equation} satisfy \begin{equation} \displaystyle\sum_{k\in \mathbb{Z}} |k||x_k|^2 <\infty. \end{equation}

If anyone knows a reference with this definition it would be perfect.

Thank you very much!