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EDIT

Oops---I found the answer to the first question of mine here on Wikipedia---this is really classic material. I'll leave the question open for a bit, in case someone tells me something interesting for my second question.

(A version of the above was mentioned by Michael at about the same time as my "oops", so I accepted his answer)

Sorry for asking a basic and naive question---if this is textbook material somewhere, please let me know, so that I may close this question.

My question is:

When is a metric space $(X,d)$ isometrically embeddable into some Banach space?

Additionally:

Can one say something "stronger", if we know that the closure of $X$ is actually a convex cone?

(By "stronger" one of the things I mean is whether we can actually obtain an embedding that is easy to compute...)

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2 Answers 2

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Arbitrarily fix $y_0\in X$. Then, with every $y\in X$ you can associate a bounded continuous function from X to R defined by $$f_y(x)=d(y,x)-d(y_0,x).$$ It is easy to show that $$\max_x |f_y(x)-f_z(x)|=d(y,z),$$ with the maximum assumed if $x=y$ or $x=z$. Hence $X$ is isometrically embedded in the Banach space $C_b(X)$.

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  • $\begingroup$ Ach, I'm having another of those "obvious after seeing the answer" moments! However, this observation surely must be a named theorem somewhere? $\endgroup$
    – Suvrit
    Commented Aug 22, 2011 at 21:44
  • $\begingroup$ Your answer answers my question and I will have to ask a more refined question to get at what I am trying! $\endgroup$
    – Suvrit
    Commented Aug 22, 2011 at 21:52
  • $\begingroup$ This is also a classical way to construct the completion of a metric space (taking its closure in the above embedding). $\endgroup$
    – Mark
    Commented Aug 23, 2011 at 12:58
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One useful embedding is into the "free space" (or "Arens-Eels space") over $X$, which is the predual of the space $Lip(X)$ of Lipschitz functions from the (pointed) metric space $(X,0)$ into the real line that map zero to zero. The free space of $X$ is just the closed span in $Lip(X)^*$ of the pointwise evaluations functionals. This millennium Godefroy and Kalton used this embedding to good effect in their paper Lipschitz-free Banach spaces, Studia Math. 159 (2003), 121–141. The embedding is described and discussed in the book N. Weaver, Lipschitz algebras, World Scientific Publishing Co. Inc., River Edge, NJ, 1999.

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  • $\begingroup$ Thanks for the nice references; Weaver seems to be advancing a viewpoint of metric spaces that I like. $\endgroup$
    – Suvrit
    Commented Aug 22, 2011 at 21:51

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