Linear Functionals and the Algebraic Conjugate
Space
Kerri Pisano
February 23, 2004
In many problems, we must associate a number with a function extracted
from a given class of functions. For example:
•to each function f(x) that has a continuous derivative on [a, b], we may
want to associate the number Rb
a(1 + [f0(x)]2)1
2dx.
•to each function f(x, y) that is twice continuously differentiable over a
closed bounded region B, we may have to form the number
R RB(∂f
∂x )2+ ( ∂f
∂y )2dxdy
•for the same function as above we associate, more simply, f(x0, y0) where
(x0, y0)∈B.
Such an association is known as a functional.
Definition 0.1 Let Xbe a linear vector space and to each xlet there be asso-
ciated a unique real (or complex) number designated by L(x). If for x, y ∈X
and for all real (or complex) α, β we have
L(αx +βy) = αL(x) + β L(y),
then Lis called a linear functional over X.
Example 0.1 X=C[a, b], i.e. the elements of Xare continous functions f(x).
We define Las
L(f) = Zb
a
f(x)dx
.
To show this is a linear functional we must verify it is a functional and it is
linear.
To see that it is a functional is easy: if f∈C[a, b] then fis continous
and therefore integrable over the interval [a, b]. Therefore the integral L(f) =
Rb
af(x)dx exists, so that Lassociates to every element fa number.
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