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

Question

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.

Answer 1

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