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Lipschitz constant of projected function

Suppose $f:\mathbb{R}^n\longrightarrow\mathbb{R}^n$ is Lipschitz with constant $K$. Suppose I project $f$ onto orthogonal subspaces of $\mathbb{R}^n$, say $X$ and $Y$, where $\mathbb{R}^n= X \oplus Y$ ...
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