  We demonstrate that spontaneous transverse polarization of Lambda baryon
($\Lambda$) production in $e^+e^-$ annihilation can be described using the
transverse momentum dependent polarizing fragmentation functions (TMD PFFs).
Using a simple Gaussian model, we perform an extraction of the TMD PFFs by
fitting the BELLE collaboration's recent measurement of the $\Lambda$
transverse polarization in back-to-back $\Lambda+h$ production in $e^+ e^-$
collisions, $e^{-} + e^{+} \rightarrow \Lambda^{\uparrow}+h+X$. We find that
this simple model accurately describes the experimental data for $\Lambda$
production associated with pions and kaons, and we are able to determine TMD
PFFs for different quark flavors. We use these newly extracted TMD PFFs to make
predictions for the transverse polarization of $\Lambda$ produced in
semi-inclusive deep inelastic scattering at a future electron-ion collider, and
find that such a polarization is around $10\%$ and should be measurable.
