Stein and Shakarchi - Functional Analysis

=Chapter 1: $$L^p$$ spaces and Banach Spaces=

The dual space of $$L^p$$ when $$1\le p<\infty$$
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.
 * 3) 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 $$\int fd\,d\nu=\int fg\,d\mu$$.
 * 4) 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 $$\nu\ll \mu$$.
 * 5) Careful: given $$\ell$$, supposing we find $$g$$ associated to it; we still have to show $||g||_q=||\ell||$. This is Lemma 4.2(ii)
 * 6) How to deal with infinite measures? Take a limit of $$g_n$$ taken from nested $$E_n$$ whose union is all $$X$$.

=Chapter 2=

Chapter 3

Chapter 4