Function Spaces What is exactly the difference between $L^2$ space and ${\ell}^1$ space? I believe that one of them is the space of square of square integrable functions. 
Does it have to do with one is for series and other for integration?
Thank You.
 A: $L^2(\Bbb R)$ is the space of square-integrable real functions:
$$L^2(\Bbb R) = \left\{ f: \Bbb R \longrightarrow \Bbb R  \mid \int_{-\infty}^\infty |f(x)|^2 \, dx < \infty \right\}.$$
Note that the above is not quite right; we say two functions $f, g \in L^2(\Bbb R)$ are equivalent if they take the same values outside of a set of measure zero.
$\ell^1(\Bbb R)$, on the other hand, is the space of absolutely convergent series:
$$\ell^1(\Bbb R) = \left\{ \{a_n\}_{n = 1}^\infty \mid \sum_{n = 1}^\infty |a_n| < \infty \right\}.$$
In general,
$$L^p(\Bbb R) = \left\{ f: \Bbb R \longrightarrow \Bbb R  \mid \int_{-\infty}^\infty |f(x)|^p \, dx < \infty \right\}$$
for $p \geq 1$ where we identify $f, g \in L^p(\Bbb R)$ if they agree outside of a set of measure zero, and
$$\ell^p(\Bbb R) = \left\{ \{a_n\}_{n = 1}^\infty \mid \sum_{n = 1}^\infty |a_n|^p < \infty \right\}.$$
If you learn measure theory at some point, you will see that there is a unifying definition of $L^p$ spaces that include the above examples as special cases: $L^p(\Bbb R)$ will be the $L^p$ space associated to $\Bbb R$ with the Lebesgue measure, and $\ell^p(\Bbb R)$ will be the $L^p$ space associated to $\Bbb N$ with the counting measure.
