A First Course in Optimization Theory

This appendix provides a brief introduction to vector spaces, and the structures (inner product, norm, metric, and topology) that can be placed on them. It also describes an abstract context for locating the results of Chapter 1 on the topological structure on ℝ n . The very nature of the material discussed here makes it impossible to be either comprehensive or complete; rather, the aim is simply to give the reader a flavor of these topics. For more detail than we provide here, and for omitted proofs, we refer the reader to the books by Bartle (1964), Munkres (1975), or Royden (1968).

Vector Spaces

A vector space over ℝ (henceforth, simply vector space) is a set V, on which are defined two operators “addition,” which specifies for each x and y in V, an element x + y in V; and “scalar multiplication,” which specifies for each a ∈ ℝ and x ∈ V, an element ax in V. These operators are required to satisfy the following axioms for all x, y, z ∈ V and a, b ∈ ℝ:

1. Addition satisfies the commutative group axioms:

1. (a) Commutativity: x + y = y + x.

2. (b) Associativity: x + (y + z) = (x + y) + z.

3. (c) Existence of zero: There is an element 0 in V such that x + 0 = x.

4. (d) Existence of additive inverse: For every x ∈ V, there is (-x) ∈ V such that x + (-x) = 0.