Page 104 - 35Linear Algebra

104 Vector Spaces

The addition is point-wise

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

as is scalar multiplication
c · f(x) = cf(x) .

R
To check that R is a vector space use the properties of addition of functions and
scalar multiplication of functions as in the previous example.
We can not write out an explicit definition for one of these functions either, there
are not only infinitely many components, but even infinitely many components between
2
any two components! You are familiar with algebraic definitions like f(x) = e x −x+5 .
However, most vectors in this vector space can not be defined algebraically. For
example, the nowhere continuous function
(
1 , x ∈ Q
f(x) = .
0 , x /∈ Q
{∗,?,#}
Example 60 R = {f : {∗, ?, #} → R}. Again, the properties of addition and
scalar multiplication of functions show that this is a vector space.

S
You can probably figure out how to show that R is vector space for any
S
set S. This might lead you to guess that all vector spaces are of the form R
for some set S. The following is a counterexample.
Example 61 Another very important example of a vector space is the space of all
differentiable functions:

d

f exists .
dx
f : R → R
From calculus, we know that the sum of any two differentiable functions is dif-
ferentiable, since the derivative distributes over addition. A scalar multiple of a func-
tion is also differentiable, since the derivative commutes with scalar multiplication
( d (cf) = c d f). The zero function is just the function such that 0(x) = 0 for ev-
dx dx
ery x. The rest of the vector space properties are inherited from addition and scalar
multiplication in R.
Similarly, the set of functions with at least k derivatives is always a vector
space, as is the space of functions with infinitely many derivatives. None of
S
these examples can be written as R for some set S. Despite our emphasis on
such examples, it is also not true that all vector spaces consist of functions.
Examples are somewhat esoteric, so we omit them.
n
Another important class of examples is vector spaces that live inside R
n
but are not themselves R .

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