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# 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 $\frac1p+\frac1q=1,1\le p-<\infty$.

Proof.

1. 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.
2. We want to go the other way: given $\ell$, find $g$; we'd like a linear functional to come from integrating against a function.
1. 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 $\int fd\,d\nu=\int fg\,d\mu$.
2. 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 $\nu\ll \mu$.
3. Careful: given $\ell$, supposing we find $g$ associated to it; we still have to show $||g||_q=||\ell||$. This is Lemma 4.2(ii)
4. How to deal with infinite measures? Take a limit of $g_n$ taken from nested $E_n$ whose union is all $X$.

Chapter 3

Chapter 4