Let $(X_1,d_1),\ldots,(X_n,d_n)$ be metric spaces and let $X=X_1\times\cdots\times X_n$ be their Cartesian product. For $x=(x_1,\ldots,x_n),\ y=(y_1,\ldots,y_n)\in X$, define $\sigma(x,y)=\sum_{k=1}^nd_k(x_k,y_k)$. Show that $\sigma$ is a metric on $X$. Show that $\sigma$ is complete if and only if $(X_i,d_i)$ is complete for each $i = 1,\ldots,n$.
Cartesian-Metric Space
0
$\begingroup$
real-analysis
metric-spaces
-
0Where did you get stuck? – 2012-12-02
-
0I could not define or show the σ on X? How can i link the σ with the Cartesian? – 2012-12-02