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Lp space

In functional analysis, the Lp 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) 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 Lp(S). For f in Lp(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 Lp space is denoted by Lp([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 Lp(R) are defined similarly.

If S is the set of natural numbers, with the counting measure, then the corresponding Lp space is denoted by l p. For p < ∞ it consists of all sequences (an) of numbers such that ∑n |an|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 Lp(S). Using the convergence theorems for the Lebesgue integral, one can then show that Lp(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 Lp for 1 < p < ∞ has a natural isomorphism with Lq where q is such that 1/p + 1/q = 1. Since this relationship is symmetric, Lp is reflexive for these values of p: the natural monomorphism from Lp to (Lp)** 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 Lp 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 Lp for 0 < p < 1 is the prototypical example of an F-space that is not locally convex[?].



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