52 CHAPTER 3. VECTOR SPACES

Now add −0x to both sides. This gives 0 = 0x. Finally,

(−1)x+ x = (−1)x+1x = (−1+1)x = 0x = 0

By the uniqueness of the additive inverse shown earlier, (−1)x =−x. ■If you are interested in considering other fields, you should have some examples other

than C, R, Q. Some of these are discussed in the following exercises. If you are happywith only considering R and C, skip these exercises. Here is an important example whichgives the typical vector space.

Example 3.0.3 Let Ω be a nonempty set and define V to be the set of functions definedon Ω. Letting a,b,c be scalars coming from a field F and f ,g,h functions, the vectoroperations are defined as

( f +g)(x) ≡ f (x)+g(x)

(a f )(x) ≡ a( f (x))

Then this is an example of a vector space. Note that the set where the functions have theirvalues can be any vector space having field of scalars F.

To verify this, check the axioms.

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

Since x is arbitrary, f +g = g+ f .

(( f +g)+h)(x)≡ ( f +g)(x)+h(x) = ( f (x)+g(x))+h(x)

= f (x)+(g(x)+h(x)) = ( f (x)+(g+h)(x)) = ( f +(g+h))(x)

and so ( f +g)+ h = f +(g+h) . Let 0 denote the function which is given by 0(x) = 0.Then this is an additive identity because

( f +0)(x) = f (x)+0(x) = f (x)

and so f +0 = f . Let − f be the function which satisfies (− f )(x)≡− f (x) . Then

( f +(− f ))(x)≡ f (x)+(− f )(x)≡ f (x)+− f (x) = 0

Hence f +(− f ) = 0.

((a+b) f )(x)≡ (a+b) f (x) = a f (x)+b f (x)≡ (a f +b f )(x)

and so (a+b) f = a f +b f .

(a( f +g))(x)≡ a( f +g)(x)≡ a( f (x)+g(x))

= a f (x)+bg(x)≡ (a f +bg)(x)

and so a( f +g) = a f +bg.

((ab) f )(x)≡ (ab) f (x) = a(b f (x))≡ (a(b f ))(x)

so (ab f ) = a(b f ). Finally (1 f )(x)≡ 1 f (x) = f (x) so 1 f = f .As above, F will be a field. It illustrates the important example of Fn, a vector space

with field of scalars F. It is a case of the above general consideration involving functions.Indeed, you simply let Ω = {1,2, · · · ,n}. We write such a function f : {1,2, · · · ,n} → Fin as an ordered list of numbers ( f (1) , · · · , f (n)). The definition, incorporating the usualnotation is as follows.