I'm struggling to understand some concepts in the first half of the first chapter of Gelfand and Fomin's Calculus of Variations. I want to make sure the defintions and theorem I wrote below are the same as the authors'.
Definition: let $C^1[a,b]$ be the collection of differentiable functions $[a,b]\to\mathbb{R}$ with continuous derivative. The space forms a vector space with the natural operations inherited from $\mathbb{R}$. Furthermore, the map $||\cdot||_1:C^1[a,b]\to\mathbb{R}$ defined by $$y\mapsto ||y||_1 := \sup_{a\le x\le b} |y(x)| + \sup_{a\le x\le b}|y'(x)|$$ is a norm on $C^1[a,b]$, providing it with the structure of a normed vector space, and thus of a topological space.
Definition: a functional $J:C^1[a,b]\to\mathbb{R}$ is continuous if it is continuous with respect to the topology provided on $C^1[a,b]$ by its norm, and the standard topology of $\mathbb{R}$. We say $J$ is differentiable at $y\in C^1[a,b]$ if its Fréchet derivative exists at that point, in which case we denote it by $\delta_y J$.
Question: are these definitions correct i.e. are they equivalent to the definitions provided in the text?
Theorem: for a fixed $C^2$ function $F:\mathbb{R}^3\to\mathbb{R}$ we define the functional $$J:C^1[a,b]\to\mathbb{R}:y\mapsto\int_b^aF(x,y,y')\ dx.$$ Then $J$ is differentiable at any $y\in C^1[a,b]$ with $$\delta_yJ:h\mapsto\int_b^a\left[F_y(x,y,y')h+F_{y'}(x,y,y')h'\right]\ dx.$$
Question: is this is the formal statement of the result stated in pages 14 and 15 of the text (image below)?
