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.



Does there exist an isometry between $L^p$ and $l^p$?
Variants of this question show up often enough here on MO and over at [math.SE](http://math.stackexchange.com) 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:



> 
> The family of Banach spaces $L^p, \ell^q$ for $1 \leq p, q \leq \infty$ consists of pairwise non-isomorphic spaces, except for two cases:
> 
> 
> 1. There is the obvious isometry between $L^2$ and $\ell^2$ and
> 2. There is a non-obvious isomorphism between $L^\infty$ and $\ell^\infty$, due to Pełczyński. However, $L^\infty$ and $\ell^\infty$ are not isometric.
> 
> 
> 


Plenty of references can be found in the following threads on MO and math.SE:


1. Bill Johnson explains in [$L^p$ vs. $L^q$](https://mathoverflow.net/questions/79713/) how one can tell these spaces apart for $1 \lt p \neq q \lt \infty$ using type and cotype considerations, and in the comments there is also some discussion of existence of embeddings.
2. A detailed explanation of the non-existence of an isomorphism between the spaces $L^p$ and $\ell^q$ for $1 \leq p,q \lt \infty$ (modulo Bill Johnson's answer in 1.) is in the answer to [If $1\leq p < \infty$ then show that $L^p([0,1])$ and $\ell\_p$ are not topologically isomorphic](https://math.stackexchange.com/q/97126). The main ingredient in that answer is Pitt's theorem stating that every operator $\ell^q \to \ell^p$ is compact for $1 \leq p \lt q \lt \infty$, the existence of an embedding $L^2 \to L^p$ plus a little bit of duality theory. See also


	* [How do you prove that $\ell^p$ is not isomorphic to $\ell^q$?](https://math.stackexchange.com/q/99086/) for a discussion how Pitt's theorem implies non-isomorphism of $\ell^p$ and $\ell^q$ and
	* [Inclusion of $L^p$-spaces, reloaded](https://math.stackexchange.com/q/170743) for a discussion of embeddings of $L^2$ into $L^p$ via Rademacher functions or Gaussians.
3. Pełczyński's isomorphism between $L^\infty$ and $\ell^\infty$ is discussed in [Is there an explicit isomorphism between $L^\infty[0,1]$ and $\ell^\infty$?](https://math.stackexchange.com/q/110438) and non-existence of an isometry is discussed in [Isometry between $L^\infty$ and $\ell^\infty$](https://math.stackexchange.com/q/207435).


Let me finish by recommending the very nice book by Albiac and Kalton, [Topics in Banach Space Theory](http://books.google.com/books?id=vjOBcgAACAAJ), as an alternative to Lindenstrauss-Tzafriri. It contains a gentle introduction to the above ideas and much more.




---


**Edit:** Further links to related topics:


* One crucial point used in establishing Pełczyński's isomorphism is the injectivity of $\ell^\infty$ and $L^\infty$. For $\ell^\infty$ this is a standard exercise in applying Hahn-Banach (coordinatewise) and for $L^\infty$ Bill Johnson gives a quick proof in [Direct proof of injectivity of $L^\infty$](https://mathoverflow.net/questions/110461/). See also [Direct proof of "K is projective iff C(K) has the Hahn-Banach property"?](https://mathoverflow.net/questions/111190/) for a generalization and related results.
* [Complemented subspaces of $\ell\_p(I)$ for uncountable $I$](https://mathoverflow.net/questions/111882/)
* [Isometric embeddings of $\ell\_q^m$ into $\ell\_p$ and $L\_p$ for $p,q\in[1,+\infty]$](https://mathoverflow.net/questions/111039/)


