The cofree coalgebra of any vector space over a field exists, though it is more complicated than one might expect by analogy with the free algebra.
The cofree coalgebra of a vector space always exists, and is unique up to canonical isomorphism.
C (V) may be constructed as a completion of the tensor coalgebra T(V) of V. For k ∈ N = {0, 1, 2, ...}, let TkV denote the k-fold tensor power of V: with T0V = F, and T1V = V. Then T(V) is the direct sum of all TkV: In addition to the graded algebra structure given by the tensor product isomorphisms TjV ⊗ TkV → Tj+kV for j, k ∈ N, T(V) has a graded coalgebra structure Δ : T(V) → T(V) ⊠ T(V) defined by extending by linearity to all of T(V).
Additional discussion of this point can be found in the tensor algebra article.
With the usual product this coproduct does not make T(V) into a bialgebra, but is instead dual to the algebra structure on T(V∗), where V∗ denotes the dual vector space of linear maps V → F. It can be turned into a bialgebra with the product
The product is dual to the coalgebra structure on T(V∗) which makes the tensor algebra a bialgebra.
Here an element of T(V) defines a linear form on T(V∗) using the nondegenerate pairings induced by evaluation, and the duality between the coproduct on T(V) and the product on T(V∗) means that This duality extends to a nondegenerate pairing where is the direct product of the tensor powers of V. (The direct sum T(V) is the subspace of the direct product for which only finitely many components are nonzero.)
However, the coproduct Δ on T(V) only extends to a linear map with values in the completed tensor product, which in this case is and contains the tensor product as a proper subspace: The completed tensor coalgebra C (V) is the largest subspace C satisfying which exists because if C1 and C2 satisfiy these conditions, then so does their sum C1 + C2.
It turns out[1] that C (V) is the subspace of all representative elements: Furthermore, by the finiteness principle for coalgebras, any f ∈ C (V) must belong to a finite-dimensional subcoalgebra of C (V).
Using the duality pairing with T(V∗), it follows that f ∈ C (V) if and only if the kernel of f on T(V∗) contains a two-sided ideal of finite codimension.
Equivalently, is the union of annihilators I 0 of finite codimension ideals I in T(V∗), which are isomorphic to the duals of the finite-dimensional algebra quotients T(V∗)/I.
The duality F[[τ]] × F[t] → F is determined by τj(tk) = δjk so that Putting t=τ−1, this is the constant term in the product of two formal Laurent series.
Conversely, any proper rational function annihilates an ideal of the form I(p).
Thus C (V) is the sum of the annihilators of the principal ideals I(p), i.e., the space of rational functions regular at zero.