# $L^p$-space on the circle, question about the definition

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 $$\mathbb{R}^{2n}$$.

• I don't understand the second question. $f$ is just a function from $S^1$ to $\mathbb R^{2n}$, and $|f(x)|^p$ is a real-valued function on $[0,1]$. – Umberto P. Apr 19 at 15:40
• @UmbertoP. Sorry for my late answer. What I don't really understand is the fact that I take a complex number $e^{2 \pi ix}$ and I should end up with an element of $\mathbb{R}^{2n}$, I mean... how do I transpose the basis $(e_n)$ if I have elements of $\mathbb{R}^{2n}$ ? – Alain Apr 19 at 21:15