Matrix Analysis

Let V be a finite-dimensional real or complex vector space with a given norm ∥·∥. The closed ball of radius ∊ about x ∈ V is B∊(x) = {y ∈ V :∥ y − x ∥≤ ∊}; the corresponding open ball is B∊ (x) = {y ∈ V :∥ y − x ∥ < ∊}. A set S ⊆ V is open if, for each x ∈ S, there is an ∊ > 0 such that B∊(x) ⊆ S. A set S ⊆ V is closed if the complement of S in V is open. A set S ⊆ V is bounded if there is an r > 0 such that S ⊆ Br(0). Equivalently, a set S ⊆ V is closed if and only if the limit of any convergent (with respect to ∥·∥) sequence of points in S is itself in S; S is bounded if and only if it is contained in some ball of finite radius. A set S ⊆ V is compact if it is both closed and bounded.

For a given set S ⊆ V and a given real-valued function f defined on S, infx∈S f(x) and supx∈S f(x) need not be finite, and even if they are, there may or may not be points xmin and xmax in S such that f(xmin) = infx∈S f(x) and f(xmax) = supx∈S f(x), that is, f need not attain a maximum or minimum value on S.