Theorems

Metric Spaces

A Best Approximation Exists

Say you have a compact subset of the metric space. For every function value in the metric space, there's a particular point of the compact subset where the distance from that point to each function value is the smallest such distance. We write

$$\forall a, b\ \exists a^*\ : d(a^*, b) \le d(a, b)\\ \text{where } a, a^* \in A, b \in B, A \sube B$$

Operator Continuity

Normed Vector Spaces

A Best Approximation Exists

Norms Have an Ordering

Convex Balls

Unique Approximations

Operator Continuity