L-p and l-p Spaces

\(L^p\) spaces and \(\ell^p\) are two related function and vector metric spaces, respectively.

\(L^p\) is the function space of functions that are square integrable:

\[L^p := \{f : \int |f(x)|^p dx < \infty \}\]

\(\ell^p\) is the vector space of functions that are square-summable:

\[\ell^p := \{v : \sum v_n^p dx < \infty \}\]

\(\ell_1\) is often called the Manhattan space, \(\ell_2\) Euclidean space.