Examples of metric spaces which are not normed linear spaces?
Give an example of a metric space which is not a normed linear space. Justify your example.
Hint: Every metric induced by a norm is unbounded.
It seems the following. Put $X=\{0;1\}$ and define a metric $d$ on the set $X$ as follows: $d(x,y)=1$ if $x\not=y$ and $d(x,x)=0$ for each $x,y\in X$. Then $X$ is not a linear space over $\mathbb R$. :-)
PS. Less trivial are examples of linear metrizable spaces admitting no consistent norm.
Hint: Consider metric $$ d:\mathbb{R}\times\mathbb{R}\to\mathbb{R}_+,(x,y)\mapsto |e^x-e^y| $$ Justfy this exmaple yourself.