Let $S$ be the space of all loops in $\mathbb{R}^2$, i.e. all continuous mappings $\gamma:[0,1]\to \mathbb{R}^2$ such that $\gamma(0)=\gamma(1)$. Is there any canonical interpretation of $S$ as a Banach space? If so, is its dual space easy to define?



The space of loops as a Banach space
The space of loops in $\mathbb R^2$ is the same as $C(\mathbb T)$, the continuous functions of the circle into the complex numbers. This is a C$^\*$-algebra and thus a Banach space.


It is well known that the dual of $C(\mathbb T)$ is $M(\mathbb T)$, the space of all Radon measures of the circle.


