Supplemental Notes for Math 704 . Spring 2012

1.1. Normed spaces. Recall that a (real) vector space V is called a normed space if there exists a function ‖ · ‖ : V → R such that (1) ‖f‖ ≥ 0 for all f ∈ V and ‖f‖ = 0 if and only if f = 0. (2) ‖af‖ = |a| ‖f‖ for all f ∈ V and all scalars a. (3) (Triangle inequality) ‖f + g‖ ≤ ‖f ||+ ‖g‖ for all f, g ∈ V . If V is a normed space, then d(f, g) = ‖f−g‖ defines a metric on V . Convergence w.r.t this metric is called norm convergence. If V is a complete metric space w.r.t. this metric, then V is called a Banach space. Let (X,B, μ) be a measure space and 0 < p ≤ ∞. Then we shall first define L(X,μ), where we will usually write L and omit X and μ. For 0 < p < ∞ we define