Let $(X,\tau)$ be a Banach space and $L\subset X$ an arbitrary finite dimensional subvector space. Let $f:L\rightarrow X$ be continuous.

1) Is the set convex hull $cx (f(L))$ also finite-dimensional?

2) If not, would it help to assume that $(X,\leq, \tau)$ is a Banach lattice and that $f$ is monotone

( i.e. $x\leq y \Rightarrow f(x)\geq f(y)$)?