Approximation Theory

Metric Spaces

• A metric space is a set such that for every $x$ and $y$ in the set there is a function $d(x,y)$ which describes the distance between them.
• A pair of objects $(A,d)$ where $A$ is a non-empty set and $d$ is a function $d:A\times A \rightarrow \mathbb{R}$ such that:
  1. $d(x,y) \geq 0$
  2. $d(x,y)=0 \Rightarrow x=y$
  3. $d(x,y) = d(y,x)$
  4. $d(x,z) \leq d(x,y) + d(y,z)$

Vector Spaces

Normed Linear Spaces

• A normed linear space is a subset of a metric space that is also a vector space.