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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?

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    $\begingroup$ Wouldn't this be $C(\mathbb T)$, continuous functions from the circle into complex numbers. It is a C$^*$-algebra and so a Banach space with dual the space of all Radon measures of $X$, $M(X)$. $\endgroup$ Feb 3, 2016 at 18:43
  • $\begingroup$ Thanks, that's exactly what I'm looking for, if you want to make this an answer I will accept it. $\endgroup$
    – Al Gale
    Feb 3, 2016 at 19:23

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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.

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