1.3.1 Function spaces, norms, and local minima

Typical function spaces that we will consider are spaces of functions from some interval to (for some ). Different spaces result from placing different requirements on the regularity of these functions. For example, we will frequently work with the function space , whose elements are -times continuously differentiable (here is an integer; for the functions are just continuous.) Relaxing the assumption, we can arrive at the spaces of piecewise continuous functions or even measurable functions (we will define these more precisely later when we need them). On the other hand, stronger regularity assumptions lead us to (smooth, or infinitely many times differentiable) functions or to real analytic functions (the latter are functions that agree with their Taylor series around every point).

We regard these function spaces as linear vector spaces over . Why are they infinite-dimensional? One way to see this is to observe that the monomials are linearly independent. Another example of an infinite set of linearly independent functions is provided by the (trigonometric) Fourier basis.

As we already mentioned, we also need to equip our function space
with a *norm*
. This is a real-valued
function on
which is
positive definite (
if
),
homogeneous (
for
all
,
), and satisfies the
triangle inequality (
). The norm gives us
the notion of a *distance*, or *metric*,
.
This allows us to define local minima and enables us to
talk about topological concepts such as convergence and continuity (more
on this in Section 1.3.4 below).
We will see how the norm plays a crucial role in the subsequent developments.

On the space , a commonly used norm is

where is the standard Euclidean norm on as before. Replacing the maximum by a supremum, we can extend the 0 -norm (1.30) to functions that are defined over an infinite interval or are not necessarily continuous. On , another natural candidate for a norm is obtained by adding the 0 -norms of and its first derivative:

This construction can be continued in the obvious way to yield the -norm on for each . The -norm can also be used on for all . There exist many other norms, such as for example the norm

where is a positive integer. In fact, the 0 -norm (1.30) is also known as the norm.

We are now ready to formally define local minima of a functional. Let
be a
vector space of functions equipped with a norm
, let
be a subset
of
, and let
be a real-valued functional defined on
(or just on
). A function
is a *local minimum* of
over
if there exists an
such that for all
satisfying
we have

Note that this definition of a local minimum is completely analogous to the one in the previous section, modulo the change of notation , , , (also, implicitly, ). Strict minima, global minima, and the corresponding notions of maxima are defined in the same way as before. We will continue to refer to minima and maxima collectively as

For the norm , we will typically use either the 0 -norm (1.30) or the -norm (1.31), with being or , respectively. In the remainder of this section we discuss some general conditions for optimality which apply to both of these norms. However, when we develop more specific results later in calculus of variations, our findings for these two cases will be quite different.