Uniform Convergence & Boundedness

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  • #1
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"Let fk be functions defined on Rn converging uniformly to a function f. IF each fk is bounded, say by Ak, THEN f is bounded."

fk converges to f uniformly =>||fk - f|| ->0 as k->∞
Also, we know|fk(x)|≤ Ak for all k, for all x

But why does this imply that f is bounded? I don't see why it is necessarily true.

Any help is appreciated!
 
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Answers and Replies

  • #2
mathman
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|f|=|f-fk+fk| ≤ |f-fk| + |fk|≤ ε + Ak

You need uniform for ε to be usable for all x.
 
  • #3
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Why does it imply f is bounded? Is ε fixed here?

Also, why do we need uniform convergence? (why is pointwise convergence not enough...I still don't see why)

Can someone explain this, please? Thanks!
 
  • #4
mathman
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Since the norm is defined to be the sup over all values of x
We can write:
||f||=||f-fk+fk|| ≤ ||f-fk|| + ||fk||≤ ε + Ak
Therefore ||f||≤ ε + Ak, which is a bound. ε will depend on k, but not on x.

The uniform convergence means that we can use one ε to be a bound for ||f-fk||. If it was simply pointwise convergence, ε would be a function of x and might not be bounded.
 
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  • #5
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Sorry, I don't think it makes sense. ε is supposed to be given, always. And we're supposed to find N such that for k>N, ... will be < ε

Also, Ak depends on fk. What we want is a bound M which is a constant and does not depend on k, right?
 
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  • #6
Landau
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You need to show that f is bounded, i.e. find some positive real number M such that for all x we have |f(x)|<M. Here M cannot depend on x, but it may ver well depend on k (why not?).

mathman found such M in his first post.
 
  • #7
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Sorry, I don't think it makes sense. ε is supposed to be given, always. And we're supposed to find N such that for k>N, ... will be < ε

Also, Ak depends on fk. What we want is a bound M which is a constant and does not depend on k, right?
put ε =1 and k = N+1
 
  • #8
mathman
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(Uniform convergence) For every ε > 0 there exists an N so that for all k > N, ||f-fk|| < ε.
(bounded) ||fk|| < Ak. (Use any k > N)

Therefore ||f|| < ε + Ak. Thus, in plain English, f is bounded!! This proof does not determine the value of the minimum bound.
 

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