I am reading a book by E. Zehnder and I am confused about an $L^p$-space he is using. Here's what is written in the book:
Start by considering integrable functions $f \in L^1(S^1)$ which are measurable functions $x \mapsto f(e^{2 \pi ix})$, periodic of period 1 and, abbreviating $f(e^{2 \pi ix}) \equiv f(x)$, satisfy
$$ \int_0^1 \mid f(x)\mid dx < \infty. $$
To each $f \in L^1(S^1)$, we will associate a sequence of numbers $ \{ \widehat{f}(j) \}_{j \in \mathbb{Z}} = \{ x_j \}_{j \in \mathbb{Z}} $ given by its Fourier coefficients defined by
$$ \widehat{f}(j) = \int_0^1 f(x)e^{-2 \pi ijx} dx. $$
The Hilbert space $L^2(S^1)$ is equipped with the scalar product
$$(f,g)= \int_0^1 f(x)\overline{g(x)} dx $$
and possesses the orthonormal system $(e_n)_{n \in \mathbb{Z}}$ defined by
$$ e_n := e^{2 \pi i nx}, \quad n \in \mathbb{Z} $$
and satisfying $(e_n,e_k)=\delta_{ik}$.
My questions :
Are those functions complex valued ? I guess so since he is using the complex exponential to define the basis.
He later uses the space $E := L^p(S^1, \mathbb{R}^{2n})$. What is the definition of this space $E$ ?
If I understand it correctly it is the set of functions from $S^1$ to $\mathbb{R}^{2n}$ which are measurable and satisfy
$$ \int_0^1 \mid f(x)\mid^p dx < \infty. $$
But this time I cannot write my functions as $x \mapsto f(e^{2 \pi i x})$ because the value should be an element of $\mathbb{R}^{2n}$.
Thank you for your help.