Liu, A note on extension of isometric embedding from a Banach space E into the universal space [l.sub.[infinity]]([GAMMA]), J.

Villa, Isometric embedding into spaces of continuous functions, Studia Math.

We should note that an interesting generalization of Mazur-Ulam theorem for isometric embeddings T : X [right arrow] Y between real normed spaces X and Y was given in [3] and then imposing an additional assumption on the range (weaker than surjectivity assumption) a Mazur-Ulamtype theorem was given in [4].

For the converse note that G[[W.sub.ba]] is isomorphic to [Q.sub.h]-1(111), thus we will first construct an isometric embedding of G[[W.sub.ba]] into [Q.sub.h]-1(111), i.e.

Convex-expansions algorithms for recognition and isometric embedding of median graphs.

Isometric embedding in products of complete graphs.

For a local and

isometric embedding to be realizable, it is necessary and sufficient that the GCRE equations hold [2,25,36,42].

Consequently there is an isometric embedding of X in [E.sup.m] and thus in [E.sup.n] also.

Theorem 3.4 and Theorem 3.12 proved in the previous section describe some necessary and sufficient conditions under which all pretangent spaces [[OMEGA].sup.X.sub.p,[??]] are isometrically embeddable in [E.sup.n] with given n but it is possible that there exists an isometric embedding of a fixed [[OMEGA].sup.X.sub.p,[??]] in [E.sup.n] even if these conditions do not occur.

Hence by Theorem 4.1 the pretangent space [[OMEGA].sup.X.sub.p,[??]] has an isometric embedding in [E.sup.n] but there are no isometric embeddings of [[OMEGA].sup.X.sub.p,[??]] in [E.sup.l] with l < n, as required.

The left completion of U([l.sup.2]) is the semigroup of all linear isometric embeddings [l.sup.2] [??] [l.sup.2] (not necessarily onto) with the strong topology.

The left completion of the group Iso(U) of isometries of the Urysohn space is the semigroup of all isometric embeddings of U into itself, with the point-wise topology.