Suppose we have a collection of sets and a set of linear functions where each being closed polyhedrons in , i.e., for some , and and . Moreover, for any , and , we have . Then we can define function on such that

We further assume is **convex**. We now state a theorem concerning the Lipschitz constant of .

**Theorem 1** * Under the above setting, given arbitrary norms , is -Lipschitz continuous with respect to and , i.e., for all , *

Here is

*Proof:* By letting , and . we see the inequality (1) is equivalent to

Using the property that is well-defined on for any , we have that there exists and corresponding such that

for all where . We also let and . These s have the property that

Thus we can bound the term by

where is due to inequality (3), is due to the definition of operator norm and is using equality (4). This proves inequality (1).

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Is there any textbook that introduce this theorem? I need a real-valued version of this theorem to cite.

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I really don’t know. Perhaps too obvious for pure mathematicians.

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