What is the definition of this $L^p$-space?

I am reading a book by E. Zehnder and I am confused about an $$L^p$$-space he is using. What is the definition of the space

$$L^p(S^1,\mathbb{R}^{2n})$$

Thank you for your kind help.

That space consists of functions $$\mathbf{f}(\theta)$$ defined on the unit circle with values in $$\mathbb{R}^{2n}$$ such that $$\int_0^{2\pi} \|\mathbf{f}(\theta)\|^p \, d\theta < \infty.$$ Here $$\| \cdot \|$$ probably (if nothing else has been said) is the ordinary norm om $$\mathbb{R}^{2n},$$ i.e. $$\| \mathbf{u} \| = \sqrt{ u_1^2 + \cdots + u_{2n}^2 }.$$
• It has to do with the Lebesgue definition of integration. In your case, when we take an interval $I \subseteq \mathbb{R}$ then we need to be able to assign a total angle to the set $\{ \theta \in [0,2\pi) : \|\mathbf{f}(\theta)\|^p \in I \},$ and this requirement is more or less the definition of measurable function. – md2perpe Apr 22 at 21:04
• Okay, I will have a look at it, thanks for your help. May I ask you a last question ? Is it true that $L^p(S^1,\mathbb{R}^{2n})=L^p(S^1,\mathbb{C}^n)$ ? I just want to use the complex exponential to define an orthonormal basis. I mean I want to use the $e_n=e^{2 \pi i nx}$ but this isn't a vector in $\mathbb{R}^{2n}$ so I thought about considering the matrix $J= \left[ {\begin{array}{cc} 0 & \mathbb{I} \\ - \mathbb{I} & 0 \\ \end{array} } \right],$ instead of the imaginary unit $i$. Is this reasoning correct ? – Alain Apr 23 at 9:40