Let $H$ be a infinite dimensional, separable Hilbert space over $\mathbb{C}$

Let $B$ a subset of $H$ such that $B$ is linearly independent and such that exists a homeomorphism $f : [0,1] \to B$ between the unit interval $[0,1]$ and $B$

I would like to know if is it true that:

For every fixed $x \in (0,1)$
$$
\overline{
\operatorname{span}
f([0,x])
}
\cap
\overline{
\operatorname{span}
f((x,1])
}
=
\operatorname{span}
f(x)
$$