Proof of the derivative of a continuous linear map

Proof

We check the definition directly: the error term is $0$.

1. Let $V$, $W$ and $A$ be as in the statement, and let $x \in V$. The domain of $A$ is $V$, which is open in $V$.
2. Let $h \in V$ with $h \ne 0$. Since $A$ is a linear map, $A(x + h) = A x + A h$.
3. So $A(x + h) - A(x) - A h = 0$, and the quotient $\|A(x + h) - A(x) - A h\| / \|h\|$ equals $0$ for each $h \ne 0$.
4. Hence this quotient tends to $0$ as $h \to 0$.
5. The map $A$ is linear and bounded by hypothesis. By the definition of the Fréchet derivative, $A$ is Fréchet differentiable at $x$ with $DA(x) = A$.
6. Since $x \in V$ was arbitrary, this holds at each point of $V$.