Let $X,Y$ be real Banach spaces.
Define $F: X \times Y \rightarrow \mathbb{R}$ as a functional mapping $\left(u,t\right) \mapsto F\left(u,t\right)$ and $F_u\left(u,t\right), F_t\left(u,t\right)$ be partial Frechet derivatives of $F.$
I want to know what are the relationships between $DF\left(u,t\right)$ and $ F_u\left(u,t\right), F_t\left(u,t\right)$.
Can you show me?
By the Riesz representation theorem, it can be represented in the form of an inner product; denoting the representing element by \begin{equation} F_u\left(u,t\right)h=\langle \text{grad}F\left(u,t\right),h \rangle \end{equation} and $$F_t\left(u,t\right)l=\langle \text{grad}F\left(u,t\right),l \rangle$$ and $$DF\left(u,t\right)w=\langle \text{grad}F\left(u,t\right),w \rangle$$.
I don't understand what $F\left(u,t\right)$ in above identities mean does?
I think that notation might be making this a little more complicated than it needs to be. This is a continual issue with vector calculus.
At every point $(u,t)$, $DF(u,t)$ is a linear map $X \times Y \rightarrow \mathbb{R}$. Similarly, $F_u(u,t)$ is a linear map $X \rightarrow \mathbb{R}$ and $F_t(u,t)$ is a linear map $Y \rightarrow \mathbb{R}$.
Because $DF(u,t)$ is a linear functional on $\mathbb{R}$, it can be identified with an element of the dual space of $X \times Y$. This is what you call the gradient of $F$ once written in dual basis coordinates. For this reason, I think that using derivatives and gradients at the same time is extremely confusing.
Just like in the one variable case, $DF(u,t)w$ measures by linear approximation the change to $F(u,t)$ when $(u,t)$ is perturbed by $w = (h,l)$. Therefore everything works out just like in the two-variable case: just as a two-variable function $f(x,y)$ has partial derivatives $f_x$ and $f_y$, and which has a full (Frechet) derivative that can be written as the matrix $\begin{bmatrix} f_x & f_y \end{bmatrix}$, the same holds for the general Banach space case.
Let $w = (h,l)$, Then the relationship becomes $$DF(u,t)w = DF(u,t)(h,l) = F_u(u,t) h + F_t(u,t) l.$$