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Exercise: Show that any linear function from $\mathbb{C}^n$ to $\mathbb{C}^m$ is Lipschitz continous. (Hint: Use suitable norms.)

I know that the maximum norm, the euclidean norm and the sum norm are all equivalent. This means that they produce the same topology, so continuity is invariant under change of these norms. But why can I chose norms when checking Lipschitz continuity which is not a topological term? Thank you in advance.

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Equivalence of norms means more than just that they induce the same topology (although, for topological vector spaces, the equality of the induced topology implies that "more"). It means that the identity $\operatorname{id} \colon (X,\lVert\,\cdot\,\rVert_1) \to (X,\lVert\,\cdot\,\rVert_2)$ is a bi-Lipschitz map.

Thus equivalent norms induce the same Lipschitz structure on the space, and hence the property of being Lipschitz-continuous is independent of the chosen representatives of the respective equivalence classes of norms on the domain and codomain.

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  • $\begingroup$ Thank you, I will try to verify this. $\endgroup$ – user114885 Sep 27 '15 at 12:16
  • $\begingroup$ $\lVert x\rVert_2 \leqslant C\cdot \lVert x\rVert_1$ is just another way to write that $\operatorname{id} \colon (X,\lVert\,\cdot\,\rVert_1) \to (X,\lVert\,\cdot\,\rVert_2)$ is Lipschitz continuous. For it also reads $$\lVert \operatorname{id}(x) - \operatorname{id}(y)\lVert_2 \leqslant C\cdot \lVert x-y\rVert_1.$$ $\endgroup$ – Daniel Fischer Sep 27 '15 at 12:20

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