Let $U, V$ be Banach spaces, and suppose $X \subseteq U$ is open. let $L(U,V)$ denote the Banach space of continuous linear operators from $U$ to $V$. Suppose that $f$ and $g$ are continuous functions in $X$ with values in $V$ and $L(U,V)$ respectively.
Suppose that there exists a set $Y \subseteq U$ such that for all $y \in Y$, whenever $x \in U, t_0 \in \mathbb{R}$, and $x + t_0y \in X$, then $t \mapsto f(x + ty)$ is differentiable at $t_0$ with derivative equal to $g(x + t_0y)y$. This of course means that that
$$\left\|\frac{f(x + (t_0 + h)y) - f(x + t_0y)}{h} - g(x + t_0y)y \right\| \to 0$$
as $h \to 0$.
I would like to show that $Y$ is a closed subspace of $U$.
This proposition is part of Theorem 1.1.6 from Hormander's The Analysis of Linear Partial Differential Operators I. I have had no trouble showing that $Y$ is closed under scalar multiplication. The trouble I am really having is showing that $Y$ is closed under vector addition. To do this, I will need to take $y_1, y_2 \in Y$, and suppose that $x + t_0(y_1 + y_2) \in X$. Then I will need to show:
$$\left\|\frac{f(x + (t_0 + h)(y_1 + y_2)) - f(x + t_0(y_1 + y_2))}{h} - g(x + t_0(y_1 + y_2))(y_1 + y_2) \right\| \to 0.$$
But I am stuck on this. I suppose I will need to add and subtract the appropriate terms, and then use the facts that $t \to f(x + ty_1)$, $t \to f(x + ty_2)$ are differentiable at $t_0$. My best attempt so far is to add and subtract $f(x + (t_0+ h)y_1 + ty_2)$. This gives
$$\frac{f(x + (t_0 + h)(y_1 + y_2)) - f(x + t_0(y_1 + y_2))}{h} =\frac{f(x + (t_0 + h)y_1 + (t_0 + h)y_2) - f(x + (t_0 + h)y_1 + t_0y_2)}{h}+ \frac{f(x + (t_0+h)y_1+ t_0y_2) - f(x +t_0y_1 + t_0y_2)}{h} $$
Using $ \tilde{x} = x + t_0y_2 \in U$, if we take $h \to 0$ in the second term, we get $g(\tilde{x} + t_0y_1)y_1$, which is a good start. The goal from here is to show that the first term converges to $g(x + t_0(y_1 + y_2))y_2$ as $h \to 0$. But I am unable to show this so far. I have tried using some of the results in Hormander's book that lead up to Theorem 1.1.6 (I have already studied them), but I am still stuck. Hints or solutions are greatly appreciated.