1.3.1 Function spaces, norms, and local minima

Typical function spaces that we will consider are spaces of functions from some interval $[a,b]$ to $\mathbb{R}^n$ (for some $n\ge 1$ ). Different spaces result from placing different requirements on the regularity of these functions. For example, we will frequently work with the function space $\mathcal C^k([a,b],\mathbb{R}^n)$ , whose elements are $k$ -times continuously differentiable (here $k\ge 0$ is an integer; for $k=0$ the functions are just continuous.) Relaxing the $\mathcal C^k$ 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 $\mathcal C^\infty$ (smooth, or infinitely many times differentiable) functions or to real analytic functions (the latter are $\mathcal C^\infty$ functions that agree with their Taylor series around every point).

We regard these function spaces as linear vector spaces over $\mathbb{R}$ . Why are they infinite-dimensional? One way to see this is to observe that the monomials $1,x,x^2,x^3,\dots$ 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 $V$ with a norm $\vert\cdot\vert$ . This is a real-valued function on $V$ which is positive definite ($\Vert y\Vert>0$ if $y\not\equiv 0$ ), homogeneous ( $\Vert\lambda y\Vert=\vert\lambda\vert\cdot \Vert y\Vert$ for all $\lambda \in \mathbb{R}$ , $y\in V$ ), and satisfies the triangle inequality ( $\Vert y+z\Vert\le \Vert y\Vert+\Vert z\Vert$ ). The norm gives us the notion of a distance, or metric, $d(y,z):=\Vert y-z\Vert$ . 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 $\mathcal C^0([a,b],\mathbb{R}^n)$ , a commonly used norm is

$$\Vert y\Vert _0:=\max_{a\le x\le b}\vert y(x)\vert$$ (1.30)

where $\vert\cdot\vert$ is the standard Euclidean norm on $\mathbb{R}^n$ 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 $\mathcal C^1([a,b],\mathbb{R}^n)$ , another natural candidate for a norm is obtained by adding the 0 -norms of $y$ and its first derivative:

$$\Vert y\Vert _1:=\max_{a\le x\le b}\vert y(x)\vert+\max_{a\le x\le b}\vert y'(x)\vert.$$ (1.31)

This construction can be continued in the obvious way to yield the $k$ -norm on $\mathcal C^k([a,b],\mathbb{R}^n)$ for each $k$ . The $k$ -norm can also be used on $\mathcal C^\ell([a,b],\mathbb{R}^n)$ for all $\ell \ge k$ . There exist many other norms, such as for example the $\mathcal L_p$ norm

$$\Vert y\Vert _{\mathcal L_p}:=\left(\int_a^b\vert y(x)\vert^p dx\right)^{1/p}$$ (1.32)

where $p$ is a positive integer. In fact, the 0 -norm (1.30) is also known as the $\mathcal L_\infty$ norm.

We are now ready to formally define local minima of a functional. Let $V$ be a vector space of functions equipped with a norm $\vert\cdot\vert$ , let $A$ be a subset of $V$ , and let $J$ be a real-valued functional defined on $V$ (or just on $A$ ). A function $y^*\in A$ is a local minimum of $J$ over $A$ if there exists an $\varepsilon >0$ such that for all $y\in A$ satisfying $\Vert y-y^*\Vert<\varepsilon$ we have

$$J(y^*)\le J(y).$$

Note that this definition of a local minimum is completely analogous to the one in the previous section, modulo the change of notation $x\mapsto y$ , $D\mapsto A$ , $f\mapsto J$ , $\vert\cdot\vert\mapsto\Vert\cdot\Vert$ (also, implicitly, $\mathbb{R}^n\mapsto V$ ). 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 extrema.

For the norm $\vert\cdot\vert$ , we will typically use either the 0 -norm (1.30) or the $1$ -norm (1.31), with $V$ being $\mathcal C^0([a,b],\mathbb{R}^n)$ or $\mathcal C^1([a,b],\mathbb{R}^n)$ , 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.