Womp 2004, Banach and Hilbert Spaces

Let X be a NLS. Consider the set of all bounded linear functionals on X, and associate the norm defined above (as an exercise verify that it is indeed a norm). The resulting space, denoted X∗, is called the dual of X. It is another exercise to verify that X∗ is a Banach space (even if X isn’t). We can also consider the double dual X∗∗. It is easy to show that X is naturally imbedded in X∗∗ by the mapping sending x ∈ X to the map taking λ to λ(x). It is another exercise to verify that this is a norm-preserving imbedding. In general, it is not true that X = X∗∗. If it is true, then X is called reflexive. However, this is true for a large number of nice spaces. You will see in first quarter analysis that for 1 ≤ p < ∞, and for “nice” measure spaces (X, dμ), that the dual of L( dμ) is L( dμ) where 1/p + 1/q = 1. It follows that L( dμ) is reflexive for 1 < p < ∞. The cases p = 1 and p = ∞ are much trickier...