Let $T$ be a subspace of the plane formed by two unitary line segments: one horizontal line,$I$, and one vertical line, $J$, which the origin is the midpoint of $I$. Show that exist an isometry $f:I \rightarrow J$, but none of the isometries $g:T \rightarrow T$ takes $I$ to $J$.
I'm really lost in this one. How can I imagine the problem and write it mathematically? I couldn't think of a way to quantify the problem.
Any help or hint is appreciated!
I'm a little confused by the question. You can just rotate the space a quarter turn about the center of line segment $I$ so that $I$ acquires the orientation of $J$. Then translate the space so that the centers line up. That's an isometry, since rotating and translating doesn't stretch anything.
But then what's the restriction about needing to map $T \rightarrow T$? If $T$ is the whole plane, then I think you can do this. Or if $T$ is a disk and the centers of the two lines coincide. If I'm understanding the question, you need some additional restriction on $T$ for the limit to be true, such as that $T$ is a nonzero proper linear subspace. In that case, the result is trivial, since then $T$ is a line, and a quarter-turn doesn't map a line to itself.