I've show that a is true. How to show that rests are false?

Question

Given that you already know that a) is true, observe that then b) and c) are equivalent, so showing either of them false closes the case. Hint: find a sequence $f_n$ so that $d_2(f_n,0) \to 0$, but $d_1(f_n,0) \geqslant 1$. — Daniel Fischer, Aug 15 '13 at 12:54
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score 0 · Accepted Answer · answered Aug 15 '13 at 12:56

Hints:

For (b): Having extremely different values on a very small range of inputs is enough to disrupt the sup norm, but not the integral norm. Think about having a very steep line on $[0,\frac{1}{n}]$, going down to 0, then being 0 on $[\frac{1}{n},1]$. How does this behave under the sup norm? Under the integral norm? It should have a limit under the integral norm that the sup norm can't agree with.

(c) You should be able to come up with this given (b).

I've show that a is true. How to show that rests are false?

1 Answers1