A vector space is a set equipped with two operations, vector addition and scalar multiplication, satisfying certain properties.

Generalizing the setup for , we have

A vector space can properly be represented as a triple , to emphasize the fact that the algebraic structure depends not just on the underlying set of vectors, but on the choice of operations representing addition and scalar multiplication.

Before proceeding to other examples, we need to discuss an important point regarding how theorems about vector spaces are typically proven. In any system of mathematics, one operates with a certain set of assumptions, called axioms, together with various results previously proven (possibly in other areas of mathematics) and which one is allowed to assume true without further verification.

In the case of vector spaces over (i.e. where the scalars are real numbers), the standing assumption is that the above list of ten properties hold for the real numbers. The fastidious reader will note that this was already assumed in the proof of Theorem thm:matalg; in fact the proof of that theorem would have been impossible without such an assumption. To illustrate how this foundational assumption applies in a different context, we consider the space Recall that i) a function is completely determined by the values it takes on the elements of its domain, and therefore ii) two functions are equal iff they have the same domain and for all elements in their common domain. So in order to show two functions and on the closed interval are equal, it suffices to verify that for all .

Next recall that the sum of two functions is defined by the equality

while the scalar multiple of the function is given by

Proof
We begin by verifying the two closure axioms. If , they are real-valued functions with common domain ; hence their sum is defined by the above equation, and has the same domain, making a function in . Similarly, if and , then multiplying by leaves the domain unchanged, so .

The eight vector space axioms [A1] - [A8] are of two types. The third and fourth are existential - they assert the existence of the zero element and additive inverses, respectively. To verify these, one simply has to produce the candidate satisfying the requisite properties. The remaining six are universal. They involve statements which hold for all collections of vectors for which the given equality makes sense. We will verify each of the eight axioms in detail. This example, then, can be used as a template for how to proceed in other cases with verification that a proposed candidate vector space is in fact one.

[A1]: For all and ,

[A2]: For all and ,

[A3]: Define by . Then for all and ,

[A4]: For each , define (note the different placement of parentheses on the two sides of the equation). Then for all and ,

[A5]: For all and ,

[A6]: For all and ,

[A7]: For all and ,

[A8]: For all and ,

Show that is a vector space, where “+” denotes matrix addition, and “” denotes scalar multiplication for matrices (in other words, verify the claim in Example example:Rmn. Hint: use the results of Theorem thm:matalg).
Let denote the set of functions on the open interval (where ), with addition and scalar multiplication defined as above in (eqn:function-add) and (eqn:function-scalar-mult). Show that is a vector space (Hint: Does the proof given above for work just as well for ?).