**EDIT**


Oops---I found the answer to the first question of mine [here on Wikipedia](http://en.wikipedia.org/wiki/Separable_space#Embedding_separable_metric_spaces)---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...)



When is a metric space isometrically embeddable into some Banach space?
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
$C\_b(X)$.


