In "Functional Analysis" by Rudin, a metric $\rho$ on the quotient space $X/N$ of a topological vector space $X$ and a closed subspace $N$ is defined as follows:
For $x,y \in X$, $ \rho (\pi(x),\pi(y)) := \inf \{d(x-y,z):z\in N\}, $ where $\pi$ is the quotient map and $d$ is an invariant metric on $X$. The verification that it is an invariant metric on $X/N$ is omitted in this book. I cannot prove the triagle inequality of the metric. Could anyone show me how to prove it ?
Thanks in advance.
Triangle inequality of a metric on a quotient space of a topological vector space
0
$\begingroup$
topological-vector-spaces
-
0@joriki, I'm sorry. X is a topological **vector** space. I corrected the question. – 2012-11-25
1 Answers
0
Let $u,v,w \in X$. $ \begin{eqnarray} \rho(\pi(u),\pi(w))&=& \inf \{ d(u-w,z) : z \in N \}, \\ &=& \inf \{ d((u-v)+(v-w),z) : z \in N \}, \\ &=& \inf \{ d((u-v)+(v-w)+z,0) : z \in N \}, \\ &=& \inf \{ d(((u-v)+z')+((v-w)+z''),0) : z',z'' \in N \},\\ &\leq& \inf \{ d((u-v)+z',0)+d((v-w)+z'',0) : z',z'' \in N \},\\ &=& \inf \{ d((u-v)+z',0): z' \in N \}+\inf \{ d((v-w)+z'',0) : z'' \in N \},\\ &=& \inf \{ d(u-v,z'): z' \in N \}+\inf \{ d(v-w,z'') : z'' \in N \},\\ &=& \rho(\pi(u),\pi(v)) + \rho(\pi(v),\pi(w)). \end{eqnarray} $
-
0inf A + inf B = inf (A+B). See for example Exercise 1.3.9. of _understanding analysis_ by Stephen Abbott. – 2014-01-13