Why we can identify $(L^p)'$ to $L^{p'}$?

Question

Let $p\in (1,\infty )$ and $(L^p)'$ the dual topological of $L^p$. Let $p'$ such that $$\frac{1}{p}+\frac{1}{p'}=1.$$ Then Riesz representation give us an isomorphism $$\Phi:L^{p'}\to (L^p)', u\longmapsto \Phi(u)$$

given by $$\Phi(u)(f)=\int uf.$$ we also have $\|u\|_{L^{p'}}=\|\Phi(u)\|_{(L^p)'}$

Indeed $\Phi$ is an isomorphism (it's also an isometry), but in what that allow us to identify $(L^p)'$ and $L^{p'}$ ? I don't really understand why we can do that. And we identify them in which sense ? They are the same in which way ?

Answer 1

If two normed linear spaces are isometrically isomorphic they have identical properties as normed linear spaces. Any theorem proved for one automatically gives validity of the same for the second space. This is the basic idea of various types of isomorphisms/homeomorphisms/iosmetric isomorphisms etc in Mathematics. Surely, the spaces are not equal as sets but they have identical properties.