Norms and Approximations

In this section we define distance in complex scalar product spaces and apply the idea to a space of functions. We show how distance lets us make precise statements about approximating functions by other functions. 

In order to exploit our intuition about distance in EucUdean geometry, we distill some of the most important properties of distance into a definition. 

Definition 3.12 Suppose V is a complex vector space and I - V — K satisfying the following  

For example, the absolute value, also known as the modulus, is a norm on 

Proposition 3.6 Suppose ( , is a complex scalar product on a complex vector space V. Define INI V R y 

---