Example 4

\* Let X be the set of all continuous mappings of the interval [0, 1] of $\mathbf{R}$ into $\mathbf{R}$. If for each pair of elements $x, y$ of X we put

then f is a pseudometric on X. \*

Remarks: 1) Example 2 above shows that a pseudometric can take the value $+\infty$ for certain pairs of elements of $X$. 2) If $f$ is a pseudometric on $X$, we can in general have $f(x, y) = 0$ without $x = y$, as Example 3 above shows (cf. § 2).

From the triangle inequality it follows that if $f(x, z)$ and $f(y, z)$ are finite then so is $f(x, y)$; moreover, in this case we have

so that (I)

If $f$ is a pseudometric on $X$, then so is $kf$ for any finite real number $k > 0$. If $\{f_i\}_{i \in I}$ is any family of pseudometrics on $X$, the sum $\sum_{i \in I} f_i$ is defined for all $x, y \in X \setminus X$; if $f$ denotes the value of this sum, then $f$ is a pseudometric on $X$. Again, the supremum of the family $\{f_i\}$ (Chapter IV, § 5, no. 5) is a pseudometric on $X$, for the relations $f_i(x, y) \leq f_i(x, z) + f_i(z, y)$ imply

[Chapter IV, § 5, no. 7, formula (17)].