Lp space

Definition

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 function if they are equal almost everywhere, we obtain the set L^p(S). For f in L^p(S), we define

<math>\|f\|_p = \left( \int |f(x)|^p \;dx \right)^{1/p}.</math>

The space L^∞(S), while related, is defined differently. We start with the set of all measurable functions from S to C (or R) which are bounded almost everywhere. By identifying two such functions if they are equal almost everywhere, we get the set L^∞(S). For f in L^∞(S), we set

<math>\|f\|_\infty = \inf \{ C\ge 0 : |f(x)| \le C \mbox{ almost everywhere} \}.</math>

Some useful special cases

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.

Further properties

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 L¹(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[?].