Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I'm looking at my old review questions from my real analysis notes from years ago, and see this problem:

Let $g$ be an integrable function. on $[0,1]$. Is there a bounded measurable function $f$ such that $$\int_{[0,1]} fg = \lVert g\rVert _1 \cdot \lVert f\rVert _{\infty}.$$

I don't know how to go about this problem, but I dug up my old notes and it seems that I have to use Riesz Representation Theorem to show the affirmative. How right or wrong am I with my hunch?

share|improve this question

1 Answer 1

Yes: take $$f(x):=1\cdot \chi_{S^+}(x)-1\cdot\chi_{S^-}(x),$$ where $S^+:=\{x,g(x)>0\}$ and $S^-:=\{x,g(x)<0\}$.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.