Consider a functional $E:C([0,1]) \rightarrow \mathbb{R}$ of the form
$$E(g) = \int_0^1 g(s)ds$$
In dealing with such functionals one often needs test functions. If one talks about the space $\mathbb{R}$ instead of $[0,1]$, then $C_0^\infty(\mathbb{R})$, the infinitely differentiable functions with compact support is suitable space of test functions. In particular one often wants to conduct integration by parts and uses that the test functions vanish at the boundary.
Now, for my example of a compact interval such an argument does not work. What would be a suitable space of test functions such that one can still use integration by parts, e.g. in deriving the Euler-Lagrange equation?