P(X, k, J) is contained in the symmetric algebra over some K-vector space, where K is a field of characteristic zero.
V denotes a finite-dimensional K-vector space of dimension r [greater than or equal to] 1 and U := [V.
Let K be a field of characteristic zero, V be a finite-dimensional K-vector space of dimension r [greater than or equal to] 1 and U = [V.
Consider K a (commutative) field and K[x] the K-vector space of polynomials in the indeterminate x.
In what follows, we generalize this result to any K-vector space with a countable basis using a pair of rather general ladder operators instead of the usual ones, namely X and D.
Let us consider a K-vector space V of countable dimension.
As a result, End(V) becomes a complete topological K-vector space (and even a complete topological K-algebra).
A simple induction using the above lemma shows that any graph containing a (k + 1)-clique is not k-vector colorable.
Further, for large k, there exists a Kneser graph K(m, r, t) that is k-vector colorable but has chromatic number exceeding [n.