CHAPTERIVNORMEDLINEARSPACESANDBANACH.PDFVIP

CHAPTER IV NORMED LINEAR SPACES AND BANACH SPACES DEFINITION A Banach space is a real normed linear space that is a complete metric space in the metric defined by its norm. A complex Banach space is a complex normed linear space that is, as a real normed linear space, a Banach space. If X is a normed linear space, x is an element of X, and δ is a positive number, then B (x) is called the ball δ of radius δ around x , and is defined by B (x) = {y ∈ X : y − x δ}. δ The closed ball Bδ (x) of radius δ around x is defined by Bδ (x) = {y ∈ X : y − x ≤ δ}. By Bδ and Bδ we shall mean the (open and closed) balls of radius δ around 0. Two normed linear spaces X and Y are isometrically isomorphic if there exists a linear isomorphism T : X → Y which is an isometry of X onto Y. In this case, T is called an isometric isomorphism. If X1 , . . . Xn are n normed linear spaces, we define a norm on the (algebraic) direct sum X = n Xi by i=1 n (x , . . . , x ) = max x . 1 n i i=1 This is frequently called the max norm. Our first order of business is to characterize those locally convex topo- logical vector spaces whose topologies are determined by a norm, i.e., those locally convex topological vector spaces that are normable. DEFINITION. Let X be a topological vector space. A subset S ⊆ 65 66 CHAPTER IV X is called bounded if for each neighborhood W of 0