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If a matrix function $g(\bf X)$ can be shown to be Lipschitz continuous in the whole space $\mathbb{R}^{N\times N}$. Then can we say that this function is also Lipschitz continuous in the real symmetric space $\mathbb{S}^N$?

Intuitively, since $\mathbb{S}^N$ is a subspace of $\mathbb{R}^{N\times N}$, this conclusion seems to be natural. However, the derivatives of symmetric matrices are different from the general matrices, which makes me a bit confused.

Can anyone give me a clear answer, or tell me if there is any reference talking about this problem? Thanks!

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