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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.

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What book were you using? –  Will Jagy Apr 18 '11 at 7:05
    
The definition you wrote is not really clear. What is an endpoint of $f$? what is $\|a\|$ if $a$ is a real number? What is the co-domain of your $f$ (I assume it is $\mathbb{R}$ or more generally some Banach space $E$). My guess is that there was some typo in the notes, and the definition was different. For instance, the space of all differentiable functions $f:\mathbb{R}\to E$ that are linear (or maybe affine) on $]M,\infty[$ and on $]-\infty, -M[$ for $M$ large enough; a suitable norm is $\|f(0)\|+\|f'\|\_{\infty}$. –  Pietro Majer Apr 18 '11 at 10:36
    
I am not sure which book the teacher was using. I'll look it up and answer when I find out. The functions $f$ in the space I am talking about are defined on intervals, so the endpoints are just the boundary of the interval. I noticed that we didn't really see a definition of "norm", it was a definition of "distance between two functions", so maybe it's only a metric space (I fixed it). We only saw examples of functions $\mathbb{R}\rightarrow \mathbb{R}$. –  Jean-Philippe Burelle Apr 18 '11 at 11:34
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