Editing Whitney embedding theorem (section)

{{short description|Any smooth real m-dimensional manifold can be smoothly embedded in real 2m-space}}
In [[mathematics]], particularly in [[differential topology]], there are two Whitney embedding theorems, named after [[Hassler Whitney]]:

*The '''strong Whitney embedding theorem''' states that any [[differentiable manifold|smooth]] [[real numbers|real]] {{mvar|m}}-[[dimension (mathematics)|dimensional]] [[manifold]] (required also to be [[Hausdorff space|Hausdorff]] and [[second-countable]]) can be [[smooth map|smoothly]] [[embedding|embedded]] in the [[real coordinate space|real {{math|2''m''}}-space]], {{tmath|\R^{2m},}} if {{math|''m'' > 0}}. This is the best linear bound on the smallest-dimensional Euclidean space that all {{mvar|m}}-dimensional manifolds embed in, as the [[real projective space]]s of dimension {{mvar|m}} cannot be embedded into real {{math|(2''m'' − 1)}}-space if {{mvar|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 {{mvar|n}}-dimensional manifold to an {{mvar|m}}-dimensional manifold may be approximated by a smooth embedding provided {{math|''m'' > 2''n''}}.  Whitney similarly proved that such a map could be approximated by an [[immersion (mathematics)|immersion]] provided {{math|''m'' > 2''n'' − 1}}. This last result is sometimes called the '''[[Whitney immersion theorem]]'''.