Vector Spaces

TL;DR: The set C(I, R) of all continuous real-valued functions on the interval I has various algebraic properties, including closure under addition and scalar multiplication, the existence of a zero function, and the existence of additive inverses.

Abstract: We saw different types of vectors last session with very similar algebraic properties. Other mathematical objects share these properties, and we will investigate these: functions, polynomials, finite vector spaces, matrices. Because they have very similar structures, techniques useful for dealing with one of these may be useful for others. Let I be an interval, for example, [0, 1], and write C(I, R) for the set of all continuous real-valued functions on I. We say that functions f and g are equal, and we write f = g, if and only if f (x) = g(x) for all x ∈ I. Given functions f and g in C(I, R) and λ ∈ R, we define new functions f + g and λf in C(I, R) as follows: (f + g)(x) = f (x) + g(x) for all x ∈ I and (λf)(x) = λf (x) for all x ∈ I. We write −f for (−1)f , that is, (−f)(x) = −f (x) for all x in I, and 0 for the zero function, i.e., 0(x) = 0 for all x in I. Proposition 1. The set C(I, R) of all continuous real-valued functions on the interval I has the following properties: (1) for all f, g ∈ C(I, R), f + g ∈ C(I, R) (closure under addition) (2) for all f ∈ C(I, R) and λ ∈ R, λf ∈ C(I, R) (closure under scalar multiplication) (3) for all f ∈ C(I, R), f + 0 = 0 + f = f (existence of zero) (4) for all f in C(I, R), f + (−f) = (−f) + f = 0 (existence of additive inverses)