Page 109 - A Course in Linear Algebra with Applications

4.1: Examples of Vector Spaces 93

Vector spaces of functions
Let a and b be fixed real numbers with a < b, and let
C[a, b] denote the set of all real-valued functions of x that are
continuous at each point of the closed interval [o, b]. If / and
g are two such functions, we define their sum f + g by the rule

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

It is a well-known result from calculus that / + g is also con-
tinuous in [a, b], so that f + g belongs to C[a, b]. Next, if c is
any real number, the function cf defined by

cf(x) = c(f(x))

is continuous in [a, b] and thus belongs to C[a,b]. The zero
function, which is identically equal to zero in [a, b], is also
included in C[a,b].
Thus once again we have a set that is closed with respect
to natural operations of addition and scalar multiplication;
C[a, b] is the vector space of all continuous functions on the
interval [a, b]. In a similar way one can form the smaller vector
space D[a, b] consisting of all differentiable functions on [a, b],
with the same rules of addition and scalar multiplication. A
still smaller vector space is £>oo[a,6], the vector space of all
functions that are infinitely differentiable in [a, b]

Vector spaces of polynomials
A (real) polynomial in an indeterminate x is an expression
of the form

f(x) = a Q + aix H \- a nx n

where the coefficients a^ are real numbers. If a n ^ 0, the
polynomial is said to have degree n. Define

Pn(R)