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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Three Squirrels 
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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