Whitney embedding theorem

In mathematics, particularly in differential topology, there are two Whitney embedding theorems, named after Hassler Whitney:

The strong Whitney embedding theorem states that any smooth real $m$ -dimensional manifold (required also to be Hausdorff and second-countable) can be smoothly embedded in the real $2 m$ -space, ⁠ $\mathbb {R} ^{2m},$ ⁠ if $m > 0$ . This is the best linear bound on the smallest-dimensional Euclidean space that all $m$ -dimensional manifolds embed in, as the real projective spaces of dimension $m$ cannot be embedded into real $(2 m - 1)$ -space if $m$ is a power of two (as can be seen from a characteristic class argument, also due to Whitney).
The weak Whitney embedding theorem states that any continuous function from an $n$ -dimensional manifold to an $m$ -dimensional manifold may be approximated by a smooth embedding provided $m > 2 n$ . Whitney similarly proved that such a map could be approximated by an immersion provided $m > 2 n - 1$ . This last result is sometimes called the Whitney immersion theorem.