Metric and Normed Spaces

Abstract

It is natural to think about distance between physical objects—people, say, or buildings or stars. In what follows, we explore — notion of “closeness” for such things as functions and sequences. (How far is f (x)= x³ from g(x) =sin x? How far is the sequence (1/n) from (2/n2)?) The way we answer such — question is through the idea of — metric space.In principle, it enables us to talk about the distance between colorsor ideas or songs. When we can measure “distance,” we can take limits or “perform analysis.” Special distance-measuring devices called norms are introduced for vector spaces. The analysis we care most about in this book involves norms. This type of analysis is known as functional analysis because the vector spaces of greatest interest are spaces of functions.