  We extend to Banach space nest algebras the theory of essential supports and
support function pairs of their bimodules, thereby obtaining Banach space
counterparts of long established results for Hilbert space nest algebras.
Namely, given a Banach space nest algebra $\mathcal A$, we charaterise the
maximal and the minimal $\mathcal A$-bimodules having a given essential support
function or support function pair. These characterisations are complete except
for the minimal $\mathcal A$-bimodule corresponding to a support function pair,
in which case we make some headway.
  We also show that the weakly closed bimodules of a Banach space nest algebra
are exactly those that are reflexive operator spaces. To this end, we crucially
prove that reflexive bimodules determine uniquely a certain class of admissible
support functions.
