Let $f:E\rightarrow F$ be a map between Banach spaces E and F; E finite dimensional (>0) and F infinite dimensional. Let $F$ be equipped with its weak topology and suppose that $f$ is strong-weak continuous.
Under what additional conditions can we guarantee that $\operatorname{span}(f(E))$ is a finite-dimensional subspace of F?
Just to give an example on how weird this can become: take $E = \mathbb{R}$ and $F = \ell^2$ with standard Hilbert basis $e_0 e_1, e_2, \ldots$. Then take a smooth bump function $\chi \in C^\infty(\mathbb{R})$ with support in the unit interval and, say $\chi(1/2) = 1$ to make things non-trivial.
Define the highly non-linear but smooth as smooth can be map \begin{equation} f(t) = \sum_{n = 0}^\infty \chi(t-n) e_n \end{equation} Then the span of the image contains the Hilbert basis...
As a slight variation one can als consider a holomorphic example, e.g. by \begin{equation} g(z) = \sum_{n=0}^\infty \frac{z^n}{n!} e_n \end{equation} Since the $e_n$ are unit vectors, this series is absolutely convergent for all $z \in \mathbb{C}$. It is holomorphic as a vector-valued function: here weakly coincides with strongly anyway, but this can also to be seen elementary. Now the derivatives at $0$ are the basis vectors again, so there is no finite-dimensional subspace and no open neighbourhood $U$ of $0$ containing $f(U)$: otherwise the derivatives would be in this subspace...
Maybe this example is a bit more illustrative since the first one has the feature that locally in $t$ the image is contained in finite-dimensional subspaces. This is not the case with $g$.
