# Extension of a non-negative and symmetric real valued function to a pseudometric

There exists a result previously stated that shows that a non-negative real valued function $$\hat{d}:X\times X\rightarrow \mathbb{R}$$ that satisfies symmetry and $$d(x,x)=0$$ (that is, different elements from the space are allowed to have distance zero) can be extended to a pseudometric? There is a result that a on the same conditions above, plus $$d(x,y)=0\Rightarrow x=y$$, then $$\hat{d}$$ can be extended to a metric.

Usually under an extension of a function $$f$$ defined on a set $$X$$ (or a pseudometric $$d$$ defined on $$X\times X$$) understood a function $$\bar f$$ defined on a set $$Y\subset X$$ (resp. a pseudometric $$\bar d$$ defined on $$Y\times Y$$), such that a restriction $$\bar f|X$$ coincides with $$f$$ (resp. $$\bar d|X\times X=f$$).

I cite (with a correction) the beginning of my student paper “On Extension of (Pseudo-)Metrics from Subgroup of Topological Group onto the Group”

“The problem of extensions of functions from subobjects to objects in various categories was considered by many authors. The classic Tietze-Urysohn theorem on extensions of functions from a closed subspace of a topological space and its generalizations belong to the known results. Hausdorff [Hau] showed that every metric from a closed subspace of a metrizable space can be extended onto the space. Isbell [4, Lemma 1.4] showed that every bounded uniformly continuous pseudometric on a subspace of a uniform space can be extended to a bounded uniformly continuous pseudometric on the whole space. The linear operators extending metrics from a closed subspace of a metrizable space onto the space were considered in, e.g., [Bes, Zar]".

If we have a symmetric non-negative function $$d$$ on $$X\times X$$ such that $$d(x,x)=0$$ for each $$x\in X$$, a standard way to modify $$d$$ to a pseudometric $$d’\le d$$ is to put

$$d’(x,y)=\inf\left\{\sum_{i=1}^{n} d(x_{i-1},x_i):x_1,\dots, x_n\in X, x_0=x, x_n=y\right\}.$$

Remark, that $$d’$$ may fail to be a metric even when $$d(x,y)=0\Rightarrow x=y$$ for each $$x,y\in X$$.

References

[Bes] Bessaga C., Functional analytic aspects of geometry. Linear extending of metrics and related problems, in: Progress of Functional Analysis, Proc. Peniscola Meeting 1990 on the 60th birthday of Professor M. Valdivia, North-Holland, Amsterdam (1992) 247-257.

[Hau] Hausdorff F., Erweiterung einer Homömorpie, - Fund. Math., 16 (1930), 353--360.

[Isb] Isbell J.R. On finite-dimensional uniform spaces, - Pacific J. of Math., 9 (1959), 107-121.

[Zar] Zarichnyi M., Regular Linear Operators Extending Metrics: a Short Proof, Bull. Pol. Ac.:Math., 44, (1996), 267--269.