Functional Analysis (Topological and Isometric Isomorphisms) Give an example that if two normed linear spaces are topologically isomorphic then they need not be isometrically isomorphic.  I searched my book and on the Internet but in vain.

Denote $X=C(K)$, $Y=C(L)$ where $K$ and $L$ are Hausdorff locally compact topological spaces.
Then by Banach-Stone theorem $X$ and $Y$ are isometric iff $K$ and $L$ are homeomorphic.
If $K$ and $L$ are both uncountable, then by Miljutin theorem $X$ and $Y$ are isomorphic.
Thus there is an abundance of examples: take any two uncountable topological spaces that are not homeomorphic and you get the desired example. Say $K=[0,1]$, $L=[0,1]\cup\{2\}$.
