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 pth 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
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
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 sigmafinite[?], 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) = (fg_{p})^{p}. The resulting metric space is complete, and L^{p} for 0 < p < 1 is the prototypical example of an Fspace that is not locally convex[?].
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