Problem: First prove that all degree-at-most-2 functions P<2 = {f(x) = ax² +bx+cla, b, c = R} is a vector space. Then find a basis for this vector space. Next, consider P to be the set of all polynomials of all degrees, can you imagine a basis set? If you can you have found your first infinite dimensional basis!!

We need to prove that all degree-at-most-2 functions $P\le 2=\left\{f\right(x)=a{x}^2+bx+c\overline{)a,b,c\in }R\}$ is a vector space.

To prove that, first, we need to know what vector space is.

Vector space:

A vector space is a set of elements, called vectors, that satisfy certain axioms. These axioms involve addition, scalar multiplication, and the properties of zero and negation.

More specifically, a vector space V over a field F consists of the following:

1. A set of elements, called vectors, denoted by V.
2. Two operations, vector addition and scalar multiplication, that satisfy the following axioms for all u, v, w in V and a, b in F:

a. u + v ∈ V (closure under addition)

b. u + v = v + u (commutativity of addition)

c. u + (v + w) = (u + v) + w (associativity of addition)

d. There exists an element 0 in V such that u + 0 = u for all u in V (existence of additive identity)

e. For each u in V, there exists an element -u in V such that u + (-u) = 0 (existence of additive inverse)

f. a(u + v) = au + av (distributivity of scalar multiplication over vector addition)

g. (a + b)u = au + bu (distributivity of scalar multiplication over field addition)

h. (ab)u = a(bu) (associativity of scalar multiplication)

i. 1u = u, where 1 is the multiplicative identity of the field F (multiplicative identity)

In other words, a vector space is a set of vectors that can be added and scaled by scalars from a field in a consistent way.