Banach Spaces which are not $L^p$ Most of the times, when I think of Banach Spaces I think of $L^p$ spaces. I would like to know if there is any Banach space which can not be written as $L^p$ space. Please also indicate any applications of such spaces. Thanks
 A: There are all sorts of weird Banach spaces.
Consider the space $c_0$ of sequences that converge to zero. This space is not (isometrically isomorphic to) an $L^p$. 
In fact, if $c_0\cong L^p$, then $\ell^1=(c_0)^*=(L^p)^*=L^{q}$, with $\frac{1}{p}+\frac{1}{q}=1$. Therefore $q=1$, because it is the only one that is not reflexive (as $\ell^1$ is not). This implies $p=\infty$. But the dual of $L^{\infty}$ is not separable (if it is infinite dimensional), while $\ell^1$ is. Therefore $c_0$ is not an $L^p$.
There are even weirder spaces. Tsirelson space, in which you can't embed an $\ell^p$, or a $c_0$. In particular, it is not an $L^p$.
James' space is isometrically isomorphic to its double dual but not reflexive. In particular it is not an $L^p$: It can't be an $L^p$, $1<p<\infty$ because those are reflexive. It is not an $L^1$ or $L^{\infty}$ because those are not isometrically isomorphic to their double duals.
A: For spaces with APPLICATIONS, you can try Sobolev spaces and things like that.
