Consider the vector space of polynomials with coefficients on a field , with the obvious sum of functions and scalar multiplication. For each , consider the subspaces spanned by polynomials of order ,
These subspaces have dimension Consider now for each the maps defined in the following way:
where if and otherwise.
Schematically, this can be written as follows
and it is not hard to prove that these maps are homeomorphisms of vector spaces over
Notice this interesting relationship between and
The kernel of and the image of are isomorphic!
The reader will surely have no trouble to show that this property is satisfied at all levels: As a consequence, for all
We say that a family of homomorphisms are boundary operators if for all If this is the case, then trivially The example above is a bit stronger, because of the isomorphism of both subspaces.
So this is the question I pose as today’s challenge:
Include a precise relationship between kernels and images of consecutive maps.
The objective of this post is to construct a pairing function, that presents us with a bijection between the set of natural numbers, and the lattice of points in the plane with non-negative integer coordinates.
We will accomplish this by creating the corresponding map (and its inverse), that takes each natural number and drops it at a location in the lattice, as the following diagram suggests: