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
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0You mention a topological vector space in the title and in the tag, but this never appears in the body of the question. – 2012-11-25
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0@joriki, I'm sorry. X is a topological **vector** space. I corrected the question. – 2012-11-25