A vector space over [math] is a set [math] of objects (called vectors) together with two operations, addition and multiplication by scalars (real numbers), that satisfy the following [math] axioms. The axioms must hold for all vectors [math] in [math] and for all scalars [math] in [math].

  1. (Closed under addition:) The sum of [math] and [math], denoted [math], is in [math].

  2. (Closed under scalar multiplication:) The scalar multiple of [math] by [math], denoted [math], is in [math].

  3. (Addition is commutative:) [math].

  4. (Addition is associative:) [math].

  5. (A zero vector exists:) There exists a vector [math] in [math] such that [math].

  6. (Additive inverses exist:) For each [math] in [math], there exists a [math] in [math] such that [math]. (We write [math].)

  7. (Scaling by [math] is the identity:) [math].

  8. (Scalar multiplication is associative): [math].

  9. (Scalar multiplication distributes over vector addition:) [math].

  10. (Scalar addition is distributive:) [math].

Let [math] be the set of functions [math]. For any two functions [math] in [math], define the sum [math] to be the function given by [math] for all real numbers [math]. For any real number [math] and any function [math] in [math], define scalar multiplication [math] by [math] for all real numbers [math].

Answer the following questions as partial verification that [math] is a vector space.

(Addition is commutative:) Let [math] and [math] be any vectors in [math]. Then [math] for all real numbers [math] since adding the real numbers [math] and [math] is a commutative operation.

(A zero vector exists:) The zero vector in [math] is the function [math] given by [math] for all [math].

(Additive inverses exist:) The additive inverse of the function [math] in [math] is a function [math] that satisfies [math] for all real numbers [math]. The additive inverse of [math] is the function [math] for all [math].

(Scalar multiplication distributes over vector addition:) If [math] is any real number and [math] and [math] are two vectors in [math], then [math]