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For each of the following, give an example of a sequence of functions $f_n(x)$ that converges to f

A. uniformly but not in the mean square sense.

B. in the mean square sense but pointwise nowhere.

I know that for part A the domain of the function cannot be bounded. Please don't just repeat the definition of what it means to converge uniformly, in the mean square sense or pointwise. I'm looking for clues on how to choose a $f_n(x)$ that matches the conditions


marked as duplicate by Ben Sheller, Daniel Fischer Apr 7 '16 at 17:41

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For each you need to choose functions that are small in one sense but large in the other. So think step functions. For a step function to be of fixed size in the square sense but nowhere point-wise you need it broad and low. For a step function to be fixed in the pointwise sense but small in the square sense you need it fixed height but narrow.

So the answer to A can be done with a sequence of wide and low boxes.

The answer to B can be done with a sequence of boxes of height 1 but skinnier and skinnier - however managing to cover every point an infinite number of times.

  • $\begingroup$ I'm a bit confused by what you mean by boxes, you talk about step function but I don't see how the two connect $\endgroup$ – Doe Apr 7 '16 at 21:31

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