Lp space
The Lp(S) spaces, for 1<=p, where S is a measure space, is the space of all Lebesgue measurable functions whose absolute value to the p'th power has a finite integral. It is a Banach space with the norm being the p'th root of the integral of the absolute value to the p'th power. Proving the triangle inequality is not completely trivial, and uses a generalization of the Cauchy-Schwarz inequality. Its dual space (the space of all continous linear functionals) has a natural isomorphism with Lq where q is such that 1/p+1/q=1. Since this relationship is symmetric, Lp is reflexive: the natural monomorphism from Lp to Lp** is onto, that is, it is an isomorphism of Banach spaces.
Some useful special cases:
- S is the measure space of the interval [0,1]. (Equivalently, S is the circle.)
- S is the natural numbers, with the measure being the number of elements in a set.