Consider the following Banach space (for concreteness):

$$X=Lip(\bar{\mathbb{B}}^n)=\{f\in C^0(\bar{\mathbb{B}}^n): \Vert f \Vert_L<\infty \}$$ where $$ \bar{\mathbb{B}}^n=\{\mathbf{x}\in \mathbb{R}^n: |\mathbf{x}|\leq 1\} $$ is the closed ball and $$ \Vert f \Vert_L=\sup_{\bar{\mathbb{B}}^n}|f| +\sup_{\mathbf{x}\neq \mathbf{y}\in \bar{\mathbb{B}}^n} \frac{|f(\mathbf{x})-f(\mathbf{y})|}{|\mathbf{x}-\mathbf{y}|} $$ is one of the usual versions of a lipschitz norm.

I'm curious what is the best way to think about the (topological) dual space $X^*$ of $X$ as this space is a bit mysterious to me.

For instance, it's clear that any finite (signed) measure on $\bar{\mathbb{B}}^n$ can be thought of as an element of $X^*$, but one should also have elements that look like differences of infinite measures whose supports are sufficiently close and whose ``relative mass" is finite. Are there other natural elements?

Any references (in particular those that are more concrete and less abstract) would be appreciated.

Edit: I guess (please correct me if I am wrong) a fairly pathological element in the dual would be something like $$ \mu=\sum_{i=1}^\infty \left( 2^i \delta_{2^{-2i}}-2^{i} \delta_{-2^{-2i}}\right) $$ on $\mathbb{B}=[-1,1]$.

Are there any natural subsets on which one can restrict and have a less crazy situation? For instance, fix a Radon measures $\mu$ on $\mathbb{B}^n$ (the open ball) with $\mu(\mathbb{B}^n)=\infty$. Is the subspace

$$Z=\{T\in X^*: T=\nu-\mu: \nu \mbox{ a Radon measure}\}$$ any nicer?

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    $\begingroup$ The space you describe contains an isomorphic copy of $l^\infty$. Its dual is wild. $\endgroup$ – Nik Weaver Jan 3 at 23:00
  • $\begingroup$ @NikWeaver Any interesting ways to tame it? For instance, I'm actually interested in something like the following: I have a fixed Radon measure $\mu$ (of infinite mass) on $\mathbb{B}^n$ (the open ball) and am interested in the set of measures $\nu$ on $\mathbb{B}^n$ so that $\nu-\mu$ lives in $X^*$. $\endgroup$ – RBega2 Jan 3 at 23:12
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    $\begingroup$ I think you want to look at the predual, which is variously known as the "Arens-Eels" or "Lipschitz-free" space. For specific information about the measures which live there see Functional Analysis by Kantorovich and Rubenstein. I don't have the reference handy but it can be found in my book on Lipschitz algebras. $\endgroup$ – Nik Weaver Jan 3 at 23:58
  • $\begingroup$ As a linear space, $Z$ is isomorphic to a subspace of Radon measures, so yes, it is nicer. I suppose you would like to say something nice about the norm on $Z$. $\endgroup$ – user95282 Jan 8 at 11:23

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