[[Differential geometry MOC]]
# Whitney embedding theorem

The **Whitney embedding theorem** establishes the equivalence of $C^\infty$ [[Differentiable manifold|differentiable manifolds]] and [[Real embedded manifold|real submanifolds]],
as well as placing an upper bound on the Euclidean dimension required to embed a given manifold.
Let $M$ be an $m$-dimensional $C^\infty$ [[differentiable manifold]].
Then $M$ is [[Diffeomorphism|diffeomorphic]] to a [[real embedded manifold]] of $\mathbb{R}^{2m}$,
i.e. $M$ may be smoothly [[Embedding|embedded]] in $\mathbb{R}^{2m}$. #m/thm/geo/diff

> [!missing]- Proof
> #missing/proof


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