I previously learned the smallest amount of calculus of variations from a physics course. I understand most part of the derivation of Euler-Lagrange equation except one or two line.
A derivation often starts by assuming that a function is a locally stationary point of a functional. What does it mean by locally stationary? Obviously we need topology to talk about "local", but it is insufficient. For example, the function $\Bbb R\to\Bbb R$, $x\mapsto x^3$ is locally stationary at $0$ by considering $\Bbb R$ as a normed space, but "locally stationary" is not defined if we consider $\Bbb R$ as purely a topological space.
Do we need some structure on the function space where we can do an analog of Taylor series expansion? I thought of this because I saw from the notes from the physics course with a sentence goes "there should be no linear terms". I would like to see some formal definitions of this "locally stationary" instead of informal arguments.