Can $L^p(\mathbb{R})$ and $ L^q(\mathbb{R})$ be isomorphic? Let $p,q \in (1,\infty)$ with $p\neq q$.  Are the Banach spaces $L^p(\mathbb{R})$, $L^q(\mathbb{R})$  isomorphic?
 A: The proof Fabian alludes to in the book reference Mark gave is a modern one using the notions of cotype and type.  One way to prove that a Banach space $X$ is not isomorphic to a Banach space $Y$ is to exhibit a property that is preserved under isomorphisms that $X$ has but $Y$ does not.  Type and cotype are examples of such properties.  The (best) type and cotype of $L_p$ are calculated in many books.  I suggest you look at Theorem 6.2.14 in the book of Albiac and Kalton.  From the statement you see that if $p\not=q$, then $L_p$ and $L_q$ either have different (best) type or different (best) cotype.
Type and cotype depend only on the collection of finite dimensional subspaces of a space (we call such a property a local property).  So you cannot use either to prove, e.g., that $L_p$ is not isomorphic to $\ell_p$ when $p\not= 2$.  One way of proving this is to show that $\ell_2$ embeds isomorphically into $L_p$ but not into $\ell_p$ when $p\not=2$.  These facts you can also find in Albiac-Kalton. You can also use infinite dimensional techniques to prove that $L_p$ and $L_q$ are not isomorphic when $p\not=q$.  Banach knew this result through infinite dimensional considerations--the concepts of type and cotype came on the scene only 40 years ago.
You will also find in Albiac-Kalton a discussion of when $L_p$ or $\ell_p$ embeds isomorphically into $L_q$. That is more complicated and in fact Banach did not know everything. He called the question the problem of the linear dimension of $L_p$ spaces, IIRC.
