I am finishing an undergraduate course on calculus of variations, and there is one thing that still annoys me about variable endpoints problems.

We learned that in a variational problem, we need to consider some "space of variations" $H$ and for the gateaux derivative to be defined it needs to be a vector space.

We did a chapter on variable endpoint problems, in which we considered the space of functions defined on an interval $[a,b]$ with $a$ and $b$ variable, and defined the following distance on this space :

$d(f,g)= \sup(|f-g|) + \sup(|f'-g'|) + ||x_0-x_0^*|| + ||x_1-x_1^*||$

where $x_0$ and $x_1$ are the boundaries of the interval of definition of $f$, and $x_0^*$ and $x_1^*$ those of $g$. The norm is the usual norm in $\mathbb{R}^2$.

We also defined the sum of two functions in this space to be the sum of their "extensions", where when you sum $f$ and $g$ you define their sum on the large interval where both are defined and extend them both linearly to the whole interval.

My question is, how is this a vector space? With addition defined as above, it seems that you can't get a function defined on a smaller interval by adding a function to another since you always "extend" the domain. We simplified matters by considering variations $h = y - y'$, meaning that we defined the variation $h$ taking $y$ to $y'$ by the difference in those two functions, but this "difference" is not compatible with the addition that we juste defined, i.e. $f + -g \neq f - g$... which is annoying.

I have tried to google this but found nothing on such a space of functions with variable endpoints.