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Suppose $M$ is a subset of a Banach space $X$.

Here, page $179$, the author defined Lipschitz-free space $\mathcal{F}(M)$ to be the canonical predual of the space of Lipschitz functions Lip$(M)$, i.e. the closed linear span of the point evaluations $$\delta_M(x)(f)=f(x), x \in M$$ in Lip$(M)^*$.

Question: I don't understand the definition of the Lipschitz-free space. Can anyone give some examples so that I can understand it?

Idonknow
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    (A rather late response, finding this from google): Chapter 10 of Ostrovskii's book 'Metric Embeddings: Bilipschitz and coarse embeddings into Banach spaces' contains a slightly different definition of the Lipschitz Free space and contains computations. If you're interested I can give more details, but his book is going to be better than me regurgitating. – James Kilbane Oct 27 '15 at 12:07
  • @JamesKilbane: I couldn't find the book on the internet. Can you give more details on it? – Idonknow Oct 27 '15 at 12:13
  • Suppose that $(X,d)$ is a metric space. We define a molecule on $X$ to be a finitely supported function such that $\sum_{x \in X} m(x) = 0$, and define the molecule $m_{pq} = 1_p - 1_q$. For a molecule define $|m|{LF} = \inf { \sum |a_i| d(p_i, q_i) : m = \sum a_i m{p_i q_i} }$. The Lipschitz Free space is then the completion of all molecules with the $|.|_{LF}$ norm. – James Kilbane Oct 27 '15 at 12:16
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    http://kaltonmemorial.missouri.edu/docs/sm2003c.pdf

    This paper (also) contains a definition of the free space, see definition 1.1. This definition just expounds on what 'canonical predual' means in this context.

     – James Kilbane Oct 27 '15 at 12:18
  • I also suggest consulting the chapter 2 from the book "Lipschitz Algebras", by Nik Weaver. There the Arens-Eells space (which is another name for the Lipschitz-free space) is presented with some geometric intuition, in connection to Lip(M). – Pedro Kaufmann Nov 27 '19 at 14:20

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