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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.

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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. $$\| \mathbf{u} \| = \sqrt{ u_1^2 + \cdots + u_{2n}^2 }.$$

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  • $\begingroup$ thanks for your comment, but should we also assume that the functions are measurable ? $\endgroup$ – Alain Apr 22 at 14:50
  • $\begingroup$ We certainly should. Otherwise the integral is not defined. $\endgroup$ – md2perpe Apr 22 at 14:54
  • $\begingroup$ okay, thank you for your comment. I am feeling quite unsure about the meaning of "measurable". May I ask you why this condition is important ? how do you understand it. Again, thank you for your help. $\endgroup$ – Alain Apr 22 at 20:48
  • $\begingroup$ 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. $\endgroup$ – md2perpe Apr 22 at 21:04
  • $\begingroup$ 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 ? $\endgroup$ – Alain Apr 23 at 9:40

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