# Linear functions $\mathbb{C}^n\longrightarrow\mathbb{C}^m$ are Lipschitz continous

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.

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.
• $\lVert x\rVert_2 \leqslant C\cdot \lVert x\rVert_1$ 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 $$\lVert \operatorname{id}(x) - \operatorname{id}(y)\lVert_2 \leqslant C\cdot \lVert x-y\rVert_1.$$ – Daniel Fischer Sep 27 '15 at 12:20