Convex hull of a continuous image

Question

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

Answer 1

Certainly not: take for instance $X:=L^2[0,1]$ and $L$ one dimensional, which we can identify with $\mathbb{R}$. Define $f(t)$ to be the characteristic function of the interval $\{s\in[0,1]: 0\le s\le t\}$.