Chapter 1: $ L^p $ spaces and Banach SpacesEdit
The dual space of $ L^p $ when $ 1\le p<\infty $Edit
Theorem 4.1: $ L^p\cong (L^q)^* $ when $ \frac1p+\frac1q=1,1\le p-<\infty $.
- We have a natural injection $ L^q\hookrightarrow (L^p)^*, g\mapsto \left(\ell:f\mapsto \int_X fg\,d\mu\right) $ by Holder's inequality. We want $ ||\ell||=||g||_q $, which is equivalent to showing that equality can be attained (or become arbitrarily close in ratio to being attained) (Lemma 4.2(i)). Just use the equality case of Holder.
- We want to go the other way: given $ \ell $, find $ g $; we'd like a linear functional to come from integrating against a function.
- The key idea is to use the Radon-Nykodim Theorem: given $ \sigma $-finite measures $ \nu\ll \mu $ (that is, $ \mu(A)=0\implies \nu(A)=0 $), there's g so that $ \int fd\,d\nu=\int fg\,d\mu $.
- How to define $ \nu $? Let $ \nu(E)=\ell(\chi_E) $. But this only works for $ E $ with finite measure, so assume first the space has finite measure. Show $ \nu $ is countably additive and $ \nu\ll \mu $.
- Careful: given $ \ell $, supposing we find $ g $ associated to it; we still have to show $||g||_q=||\ell||$. This is Lemma 4.2(ii)
- How to deal with infinite measures? Take a limit of $ g_n $ taken from nested $ E_n $ whose union is all $ X $.