Derivative of functional - how is it defined? I am reading these notes on calculus of variations by Joel G. Broida from the University of Colorado. 
On pg. 2 the differential ($L$) of a functional ($F$) is defined as follows:
$$F[\gamma+h]-F[\gamma]=L(h)+R(h,\gamma)$$
where $\gamma(t)$ is the curve the extremizes $F$, $h(t)$ is an arbitrarily small function (loosely stated...) and $R$ is a remainder with the property that $R(h,\gamma)=\mathcal{O}(h^2)$. In order for $F$ to be differentiable $L$ must be a linear map.
My question: are $L$ and $R$ functionals as well? 
The reason I ask is because their argument is $h(t)$,  which is a function.
If so, what is the meaning of $R(h,\gamma)=\mathcal{O}(h^2)$ for a functinonal?
What does it mean that $h(t)\rightarrow0$ without a change in $t$?
 A: Like in multivariable calculus, one can define both a directional derivative and a derivative or differential of a functional.  The distinction is as follows: 
The directional derivative (or variation) of the functional $F[x]$ ($x$ is an element of a Banach space, say $X$) is the following limit, if it is defined: 
$$
\delta F[x;y] = \lim_{\epsilon\rightarrow 0}\frac{F[x+\epsilon y]-F[x]}{\epsilon}
$$  For each fixed $x$, the variation is a functional of $y$ i.e. it eats a direction $y$ and returns a number $\delta F[x;y]$
As you say, in order that $F$ be truly differentiable, we require slightly more than the directional derivative to exist for each $y$: we require a linear approximation to $F$.  Since $F$ is a functional, a linear approximation to $F$ will take the form of a linear functional.  From Riesz representation, this is equivalent to the existence of a fixed function $F^\prime[x]\in X^\prime$, called the functional derivative, such that the following holds: 
$$
\delta F[x;y] = \langle F^\prime[x],y\rangle
$$ 
The existence of a functional derivative is equivalent to $F$ being differentiable.  To summarize: 


*

*For each $x$, the directional derivative is a functional $\delta F[x;\cdot]$

*If this functional is linear in $y$, there exists a representative vector $F^\prime[x]$ for which the directional derivative can be computed via the formula above.


This is exactly the same situation as multivariable calculus, where the directional derivative can be computed as a dot product between the gradient vector and the unit direction; the functional derivative plays the role of the gradient vector.
As for your second question, orders of convergence for functionals require the definition of a norm.  So saying a functional is $o(h^2)$ is really saying it's $o(\|h\|^2)$ where $\|\cdot\|$ is the appropriate norm (usually a Banach or Hilbert space norm). 
