The motivation is simple, as it is trivially right when $p=2$. When considering the duality between $L^p$ ($l^p$) and $L^q$ ($l^q$) when $p$ and $q$ are conjugate in the sense that $1/p+1/q=1$, I wonder if $L^p$ and $l^p$ are the same in the sense of isometry. I tried to use the situation when $p=2$, however I find it difficult to give a linear homeomorphism.

Variants of this question show up often enough here on MO and over at math.SE that it seems worthwhile to collect some facts and links. I say isomorphic for linearly homeomorphic and isometric for isometrically isomorphic. One main upshot is:
Plenty of references can be found in the following threads on MO and math.SE:
Let me finish by recommending the very nice book by Albiac and Kalton, Topics in Banach Space Theory, as an alternative to LindenstraussTzafriri. It contains a gentle introduction to the above ideas and much more. Edit: Further links to related topics:



The standard way to do this is to use the fact that infinite dimensional complemented subspaces of $\ell^p$ are isomorphic to the space itself whereas $L^p$ has a complemented subspace which is isomorphic to Hilbert space (essentially the Khintchine inequality). Probably the most accessible reference is the standard monograph of Lindenstrauss and Tzafriri on classical Banach spaces. The $\infty$ case is a bit more subtle. 

