L p-SPACES FOR 0 < p < 1

In a first course in functional analysis, a great deal of time is spent with Banach spaces, especially the interaction between such spaces and their dual spaces. Banach spaces are a special type of topological vector space, and there are important topological vector spaces which do not lie in the Banach category, such as the Schwartz spaces. The most fundamental theorem about Banach spaces is the Hahn-Banach theorem, which links the original Banach space with its dual space. What we want to illustrate here is a wide collection of topological vector spaces where the Hahn-Banach theorem has no obvious extension because the dual space is zero. The model for a topological vector space with zero dual space will be Lp[0, 1] when 0 < p < 1. After proving the dual of this space is {0}, we’ll see how to make the proof work for other Lp-spaces, with 0 < p < 1. The argument eventually culminates in a pretty theorem from measure theory (Theorem 4.2) which can be understood at the level of a first course on measures and integration.