Let $T :\mathbb{R}^m \rightarrow \mathbb{R}^n $ be a linear map, where $m > n$. Then, it is possible to prove that there is no $T$ injective in this case. In fact, if such a map exists the Rank-Nullity Theorem gives us $$ n \geq \dim Im (T) = m > n, $$ so, we obtain a contradiction. A natural question arises here:
It is possible to obtain a injective continuous (nonlinear) map $f:\mathbb{R}^m \rightarrow \mathbb{R}^n$ when $m > n$ ?
I try to construct such an example, but it seems so delicate. There exist some way to prove that a such map does not exist ?
Suppose there was such an $f$, so $f$ is an embedding $\mathbb{R}^m \to \mathbb{R}^n$. Then since $m>n$, there is a copy of $S^n$ inside $\mathbb{R}^m$, and $f$ must restrict to an embedding $S^n \to \mathbb{R}^n$.
But such maps don't exist because $S^n$ can't be embedded in $\mathbb{R}^n$.