Let $(V, \|\cdot\|)$ be a normed vector space and let $K\subset V$ compact and convex. I want to show that there exists a continuous surjection $f:[0,1]\to K$ (that is, $f([0,1])$ is a space-filling curve).
We know that there exists a continuous surjection from the Cantor set $C$ to $K$, call it $g: C\to K$. Now, $[0,1] \setminus C$ is a countable union of open intervals $(a,b)$. Pick one such interval and define, for $t\in (a,b)$,
$$\phi((1-t)a+bt)=(1-t)\phi(a)+t \phi(b)$$
Then, if we define $f(x)=\cases{g(x) & if $x\in C$\\ \phi(x) & otherwise}$
Do you think this might work? I'm not entirely convinced that $\phi$ will really do the job, since I'm not sure if $a$ and $b$ will actually get mapped to $K$. Would appreciate some advice.
Your construction works, but you should write $$\phi((1-t)a+bt)=(1-t)g(a)+t g(b)$$ (with $g$ on the right hand side).
Since $C$ is closed, the endpoints $a,b$ belong to $C$. And since $g$ maps $C$ into $K$, we have $g(a), g(b)\in K$, hence $(1-t)g(a) + t g(b)\in K$.
The map $f$ could be described as "piecewise linear extension of $g$".