Let V be the set of functions f:R→R. For any two functions f, g in V, define the sum f+g to be the function given by (f+g) (x) = f (x) + g (x) for all real numbers x. For any real number c and any function f in V, define scalar multiplication cf by (cf) (x) = cf (x) for all real numbers x.

To check that V is a vector space it suffice to show

1. Associativity of vector addition.

2. Additive identity

3. Existence of additive identity

4. Associativity of scalar multiplication

5. Distributivity of scalar sums

6. Distributivity of vector sums

7. Existence of scalar multiplication identity.

Step-by-step explanation:

To see that V is a vector space we have to see that.

1. Associativity of vector addition.

This property is inherited from associativity of the sum on the real numbers.

2. Additive identity.

The additive identity in this case, would be the null function f (x) = 0. for every real x. It is inherited from the real numbers that the null function will be the additive identity.

3. Existence of additive inverse for any function f (x).

For any function f (x), the function - f (x) will be the additive inverse. It is in inherited from the real numbers that f (x) - f (x) = 0.

4. Associativity of scalar multiplication.

Associativity of scalar multiplication is inherited from associativity of the real numbers

5. Distributivity of scalar sums:

Given any two scalars r, s and a function f, it will be inherited from the distributivity of the real numbers that

(r+s) f (x) = rf (x) + sf (x)

Therefore, distributivity of scalar sums is valid.

5. Distributivity of vector sums:

Given scalars r and two functions f, g, it will be inherited from the distributivity of the real numbers that

r (f (x) + g (x)) = r f (x) + r g (x)

Therefore, distributivity of vector sums is valid.

6. Scalar multiplication identity.

The scalar 1 is the scalar multiplication identity.