What is the derivative of the following functional?

Question

How can we find the derivative of the following functional w.r.t the function $\lambda$:

\begin{equation*} \mathcal{J}(\lambda) = \int_0^1 \left( \int_t^1 \lambda(s) ds \right)dt \end{equation*}

where $\lambda \in L_2[0,1]$.

I guess we should apply Gateaux differential:

\begin{equation*} \lim_{\alpha \to 0} \frac{1}{\alpha} \mathcal{J}(\lambda +\alpha h) - \mathcal{J}(\lambda) \end{equation*}

with $h\in L_2$ being arbitrary. or?

Answer 1

The functional $\mathcal{J}$ is linear hence $\mathcal{J}'=\mathcal{J} .$

Answer 2

If $X,Y$ are normed spaces and $T:X\to Y$ is a linear operator then $T' =T. $ This follows immadietely from equality $$\frac{||T(x+h)-T(x)-T(h)||}{||h||} =0$$