However, the discrete metric space is free in the category of bounded metric spaces and Lipschitz continuous maps, and it is free in the category of metric spaces bounded by 1 and short maps.
2.
That is, any function from a discrete metric space to another bounded metric space is Lipschitz continuous, and any function from a discrete metric space to another metric space bounded by 1 is short.
3.
The "'Kuratowski WojdysBawski "'theorem states that every bounded metric space " X " is isometric to a convex subset of some Banach space . ( N . B . the image of this embedding is closed in the convex subset, not necessarily in the Banach space . ) Here we use the isometry
4.
Furthermore, \ delta _ X \ colon x \ mapsto d _ x, where d _ x ( p ) : = d ( x, p ) \,, is an isometric embedding of X into \ operatorname { Aim } ( X ); this is essentially a generalisation of the Kuratowski-WojdysBawski embedding of bounded metric spaces X into C ( X ), where we here consider arbitrary metric spaces ( bounded or unbounded ).
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