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.

  • $\begingroup$ 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 $[0,1]$. $\endgroup$ – Umberto P. Apr 19 at 15:40
  • $\begingroup$ @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 $\mathbb{R}^{2n}$ ? $\endgroup$ – Alain Apr 19 at 21:15

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