I'm trying to understand the relationship between two different definitions of differentiability for a map $f: \mathbf{V} \to \mathbf{W}$, where $\mathbf{V}$ and $\mathbf{W}$ are topological vector spaces.
Differentiability via compositions: A map $f:\mathbf{V}\to\textbf{W}$ is differentiable if for every smooth function $\phi: \mathbf{W} \to \mathbb{R}$, the composition $\phi\circ f$ is smooth (differentiable).
Directional Derivative: the directional derivative of $f$ at $\textbf{x}$ in the direction $\textbf{v}$ exists (for all $\textbf{x}$ and $\textbf{v}$), that is, the limit $$ D_{\mathbf{v}} f(\mathbf{x}) = \lim_{h \to 0} \frac{f(\mathbf{x} + h \mathbf{v}) - f(\mathbf{x})}{h} $$ exists.
Question: Does the first definition imply the second?
In other words, if for all smooth real-valued functions $\phi$ the composition $\phi\circ f$ is differentiable, can we conclude that $f$ has directional derivatives?
In finite dimensions we can, assuming $f$ is differentiable in the first sense, show that the coordinate functions of $f$ are differentiable, thus yielding differentiability of $f$ as in the second definition. It isn't clear to me how (or if) this may be extended to arbitrary TVSs.
Any clarification or insight would be greatly appreciated! Thanks!
