Partial Derivative of Frechet-differentiable Function

Question

Let $X,Y,Z$ be Banach-Spaces and $F:X \times Y \rightarrow Z$ Frechet-differentiable. Then it holds $$ F'(x,y)(u,v) = F_x (x,y)u+ F_y(x,y)v .$$

How do I prove this?

Answer 1

Define projections $p_1(x,y) = x_1$ and $p_2(x,y) = y$ and injections $j_1(x) = (x, 0)$ and $j_2(y) = (0, y).$ Then, $I_{\mathrm{X} \times \mathrm{Y}} = j_1 \circ p_1 + j_2 \circ p_2.$ The result follows by definition of partial derivatives and usual differentiation rules upon noticing $F = F \circ I_{\mathrm{X} \times \mathrm{Y}}.$