Lp space (original) (raw)

In functional analysis, the L_p_ spaces form an important class of examples of Banach spaces and topological vector spaces.

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) whoseabsolute 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 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

Some useful special cases

The most important case is when p = 2, the space L2 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.

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.