Table of contents |

2 Some useful special cases 3 Further properties |

We start with a positive real number *p* and a measure space *S* and consider the set of all measurable functions from *S* to **C** (or **R**) whose
absolute value to the *p*-th power has a finite Lebesgue integral. Identifying two such functions if they are equal almost everywhere, we obtain the set L^{p}(*S*).
For *f* in L^{p}(*S*), we define

The most important case is when *p* = 2, the space L^{2} is a
Hilbert space, that has important applications to
Fourier series and quantum mechanics

If one chooses *S* to be the unit interval [0,1] with the Lebesgue measure, then the corresponding L^{p} space is denoted by L^{p}([0,1]). For *p* < ∞ it consists of all functions *f* : [0,1] → **C** (or **R**) so that |*f*|^{p} has a finite integral, again with functions that are equal almost everywhere being identified. The space L^{∞}([0,1]) consists of all measurable functions *f* : [0,1] → **C** (or **R**) such that |*f*| is bounded almost everywhere, with functions that are equal almost everywhere being identified. The spaces L^{p}(**R**) are defined similarly.

If *S* is the set of natural numbers, with the counting measure, then the corresponding L^{p} space is denoted by *l*^{ p}. For *p* < ∞ it consists of all sequences (*a*_{n}) of numbers such that ∑_{n} |*a*_{n}|^{p} is finite. The space *l*^{∞} is the set of all bounded sequences.

If 1 ≤ *p* ≤ ∞, then the Minkowski inequality, proved using Hölder's inequality, establishes the triangle inequality in L^{p}(*S*).
Using the convergence theorems for the Lebesgue integral, one can then show that L^{p}(*S*) is complete and hence a Banach space.
(Here it is crucial that the Lebesgue integral is employed, and not the Riemann integral.)

The dual space (the
space of all continuous linear functionals) of L^{p} for 1 < *p* < ∞ has a natural isomorphism with
L^{q} where *q* is such that 1/*p* + 1/*q* = 1. Since this relationship is
symmetric, L^{p} is reflexive for these values of *p*:
the natural monomorphism from L^{p} to (L^{p})^{**} is
onto, that is, it is an isomorphism of Banach spaces. If the measure on *S* is sigma-finite, then the dual of L^{1}(*S*) is isomorphic to L^{∞}(*S*).

If 0 < *p* < 1, then L^{p} can be defined as above, but it won't be a Banach space as the triangle inequality does not hold in general. However, we can still define a metric by setting *d*(*f*,*g*) = (||*f*-*g*||_{p})^{p}. The resulting metric space is complete, and L^{p} for 0 < *p* < 1 is the prototypical example of an F-space that is not locally convex.