Show that something is Fréchet differentiable

Question

I have that $Y$ is a complete normed linear space and denote $M=\mathcal{L}(Y)$, the space of linear operators $A:Y\rightarrow Y$. Also, let $F:M\rightarrow M$ be the map defined by $F(A)=A^2$. How can I show that F is Fréchet differentiable at each $A \in M$? Thanks in advance for any guidance!

Answer 1

$(A+h)^2 = A^2 +hA+ Ah +h^2$

$h \to hA+Ah$ is continuous linear and $h^2 = o(\Vert h \Vert)$.

Conclusion: $F$ is Fréchet differentiable and $F^\prime(A)(h) = hA+Ah$.