[chongli]

You have to study quite a lot of linear algebra before you can tackle functional analysis, the subject which includes the study of topological vector spaces. I don't know anything about robotics, however, so I don't know how functional analysis would apply, if at all.

[tnecniv]

Functional analysis shows up all over the place. Everywhere you can take a Fourier transform, you can use functional analysis. For robotics, control theory and computer vision both come to mind.