I'm reading about the isometric embedding and the definition goes like this. Let $(X, d)$ and $(Y, d_1)$ be metric spaces and $f$ a mapping of $X$ into $Y$. Let $Z=f(X)$, and $d_2$ be the metric induced on Z by $d_1$. If $f: (X, d) → (Z, d_2)$ is an isometry, then $f$ is said to be an isometric embedding of $(X, d)$ in $(Y, d_1)$. I'm not able to understand the metric induced by another metric part. Can someone give an example so that I can understand this?
A metric induced by another metric
metric-spaces
Best Answer
The definition is $d_2(f(x),f(y))=d_1(x,y)$. For this to be a metric we have to assume that $f$ is injective.
For example let $f:\mathbb R \to \mathbb R$ be defined by $f(x)=e^{x}$. the range of this function is $(0,\infty)$ and the induced metric $d_2$ on the range is defined by $d_2(x,y)=|\log \,x -\log \, y |$ (assuming that the domain of $f$ is given the usual metric).