By a functional we mean a mapping from some space of curves to $\mathbb{R}$. A functional can have infinitely many extremals or none. If $F$ is the length of curve connecting the north and south pole of some 3D ball, then $F$ has infinitely many extremals. I can't find an example where some functional $F$ has no extremal.
2026-04-04 00:54:27.1775264067
Extremal of functionals
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Consider the space $C([0,1]; \mathbb{R})$ of continuous curves on the real line. Consider the functional: $$I(x) = \begin{cases} ||x||_{\infty} \text{ for } x \neq 0 \\ 1 \text{ for } x = 0\end{cases}$$
This functional has no minimizer and no maximizer. The notable things are:
this functional has a finite infimum.
this functional has a infimizing converging sequence, but the point of convergence is not a minimizer. This is because of lack of lower semicontinuity.