Separation in General Normed Vector Spaces

The goal here is to extend this result to general normed vector spaces over the reals. Informally, a “vector space over the reals,” is a set X together with the operations vector addition and scalar multiplication, which are assumed to have the usual properties (e.g., vector addition is commutative). “Over the reals” means that scalars are in R rather than in some other field, such as the complex numbers. Henceforth, I write “vector space” rather than “vector space over the reals.” A norm on X is a function, denoted ‖ · ‖, from X to R+ such that for all x, x̂ ∈ X and for all θ ∈ R, 1. ‖x‖ = 0 iff x = 0, 2. θ ∈ R, ‖θx‖ = |θ|‖x‖, 3. ‖x+ x̂‖ ≤ ‖x‖+ ‖x̂‖. Examples of normed vector spaces, in addition to RN , include the following. • `∞. The set of points in R∞ such that ‖x‖sup = sup n |xn| <∞.