Hard Whitney embedding theorem

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This article is about an embedding theorem, viz about sufficient conditions for a given manifold (with some additional structure) to be realized as an embedded submanifold of a standard space (real or complex projective or affine space)
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Statement

A compact connected differential manifold of dimension n can be embedded inside \R^{2n}.

This is an improvement on the Whitney embedding theorem, which states that any compact connected differential manifold of dimension n can be embedded inside \R^{2n+1} and immersed inside \R^{2n}.