In our lecture there is the following sentence:
"The function $J:H^1_{a}(a,b) \times L^\infty(a,b) \to \mathbb{R}$ is assumed to be Fréchet differentiable. We assume the partial derivatives can be extended to linear and continous functionals on $L^2(a,b)$ for which we will be using the same symbol. For example $\frac{\partial J}{\partial u}(\bar{y}, \bar{u}) \in L(L^2(a,b), \mathbb{R}) = L^2(a,b)$."
I do not understand the last two sentences. For me the Fréchet derivative $\frac{\partial J}{\partial u}(\bar{y}, \bar{u})$ is a linear and continous function from $L^\infty(a,b)$ to $\mathbb{R}$ hence $\frac{\partial J}{\partial u}(\bar{y}, \bar{u}) \in L(L^\infty(a,b), \mathbb{R})$. I understand that this function can be interpreted as a function in $L(L^2(a,b), \mathbb{R})$ by the assumption. But how could you describe (and treat) this as an element in $L^2(a,b)$?