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发帖数: 2039 | 1 We have a set of continuous functions, F={f∣f:A→R}, where the fixed domain
A
is compact in R^n. Now if for some given constant t∈A, and any e>0, there
exists f in F such that max_A f_e-f_e(t)
exists
f∈F, such that f(t)=max_A f? The uncertain part is that you could assume F
to
satisfy some regular conditions, like compact wrt some standard topology,
convex, and so on. However, it is for sure that the less restrictive the
imposed conditions are the better.
Any relevant results or theorems would help. Many thanks! | E*****T
发帖数: 1193 | 2 I guess the difficulty is that the space of continous function with sup norm
is not reflexive. You can try to prove it with additional condition: F
closed, convex (maybe also bounded?) I don't know if this is right.
But if you let F be a closed, convex (bounded?) subset of some Sobolev space
which can be embedded into C^0, then a minimizer should exists, since
Sobolev space is reflexive. |
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