The set of L^p-functions (where p>=1) generalizes L^2-space. Instead of square integrable, the measurable function f must be p-integrable for f to be in L^p. On a measure space X, the L^p norm of a function f is left bracketing bar f right bracketing bar _L^p = ( integral_X ( left bracketing bar f right bracketing bar )^p)^(1/p). The L^p-functions are the functions for which this integral converges. For p!=2, the space of L^p-functions is a Banach space which is not a Hilbert space.

Banach space | completion | Hilbert space | L^2-space | Lebesgue integral | measure | measure space