Show that $\|F\|=\|f\|_q$.

Question

Let $L^q$ be the space of $L^q$-integrable functions on the complete measure space $(X,B,\mu)$. Let $f$ be a given function in $L^q$. Define $F$ on $L^p$ by $F(g)=\int fg\,d\mu$. Show that $\|F\|=\|f\|_q$.

$$|F(g)|=\left|\int fgd\mu\right|\le\int |fg|d\mu\le\|f\|_q\|g\|_p$$ by Holder's inequality. So we have $\|F\|\le\|f\|_q$. But how do you show that $\|F\|\ge\|f\|_q$?

Maybe you should think about the Equality case in Hölder's and Minkowski's inequality — Martin, Jan 05 '13 at 06:34

Koushik · Accepted Answer · 2013-01-05T07:17:10.410

0

this is called duality principle for L^P spaces.If you have royden's real analysis you may look at the solution or look at this place lemma 4.2 http://press.princeton.edu/chapters/s9627.pdf the main idea is to construct a proper g of norm 1

edited Jan 05 '13 at 07:17

answered Jan 05 '13 at 07:01

Koushik

4,472

Show that $\|F\|=\|f\|_q$.

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