Ok. “Rigged Hilbert space” and “Complete Lattice” connection:
Rigged Hilbert spaces are a relevant example of partial inner product (Pip-) spaces. A Pip-space V is characterized by the fact that the inner product is defined only for compatible pairs of elements of the space V. It contains a complete lattice of subspaces (the so-called assaying subspaces) fully determined by the compatibility relation. An assaying subspace is nothing but an interspace, in the terminology adopted here. The point of view here is however different: we start from a RHS and look for convenient families of interspaces for which certain properties are satisfied. Nevertheless, we believe that an analysis similar to that undertaken here could also be performed in the more general framework of Pip-spaces, but this problem will not be considered here.
I didn’t know about Mathematical Morphology nor about complete lattices in 2013.
Those I learned in 2018 or 2019.
Now in 2020, I’m learning that “Rigged Hilbert space” is not only formed of complete lattices (which means I could work with mathematical morphology if I wanted to) – but ALSO that it can be generalized in a family of “Partial inn⍺er product” (PIP) spaces.
So now I have to learn what PIP spaces are but I think it means that “it’s ok to ignore the parts that don’t match up perfectly so long as your heart’s in the right place”. In short, close enough to be treated as-if continuous, allowing you to carry on without nailing down every detail perfectly because it’s probably impossible to make perfect pairs anyway.
But, within that interspace there is a lot of flexibility.
quote came from: https://www.sciencedirect.com/science/article/pii/S0022247X13009293
The duals of metaphors are inner part of the Venn diagram where there’s matching bits.
Now, turn that Venn diagram inside out and you have rigged hilbert space where the non-matching bits of the metaphors reside – in the interspace.