dual of a coalgebra is an algebra, the
Let $R$ be a commutative ring with unity. Suppose we have a coassociative coalgebra $(C,\mathrm{\Delta})$ and an associative algebra $A$, both over $R$. Since $C$ and $A$ are both $R$modules, it follows that ${\mathrm{Hom}}_{R}(C,A)$ is also an $R$module. But in fact we can give it the structure of an associative $R$algebra^{}. To do this, we use the convolution product^{}. Namely, given morphisms $f$ and $g$ in ${\mathrm{Hom}}_{R}(C,A)$, we define their product $fg$ by
$$(fg)(x)=\sum _{x}f({x}_{(1)})\cdot g({x}_{(2)}),$$ 
where we use the Sweedler notation
$$\mathrm{\Delta}(x)=\sum _{x}{x}_{(1)}\otimes {x}_{(2)}$$ 
for the comultiplication $\mathrm{\Delta}$. To see that the convolution product is associative, suppose $f$, $g$, and $h$ are in ${\mathrm{Hom}}_{R}(C,A)$. By applying the coassociativity of $\mathrm{\Delta}$, we may write
$$((fg)h)(x)=\sum _{x}(f({x}_{(1)})\cdot g({x}_{(2)}))\cdot h({x}_{(3)})$$ 
and
$$(f(gh))(x)=\sum _{x}f({x}_{(1)})\cdot (g({x}_{(2)}))\cdot h({x}_{(3)}).$$ 
Since $A$ has an associative product, it follows that $(fg)h=f(gh)$.
In the foregoing, we have not assumed that $C$ is counitary or that $A$ is unitary. If $C$ is counitary with counit $\epsilon :C\to R$ and $A$ is unitary with identity $1:R\to A$, then their composition^{} $1\circ \epsilon :C\to A$ is the identity for the convolution product.
Example.
Let $C$ be a coassociative coalgebra over $R$. Then $R$ itself is an associative $R$algebra. The algebra ${\mathrm{Hom}}_{R}(C,R)$ is called the algebra dual to the coalgebra $C$.
We have seen that any coalgebra dualizes to give an algebra. One might expect that a similar construction could be performed on ${\mathrm{Hom}}_{R}(A,R)$ to give a coalgebra dual to $A$. However, this is not the case. Thus coalgebras (based on “factoring”) are more fundamental than algebras (based on “multiplying”).
(The proof will be provided at a later stage).
Remark on Al/gebraic Duality–Mirror or tangled ‘duality’ of algebras and ‘gebras’:
An interesting twist to duality was provided in Fauser’s publications on al/gebras
where mirror or tangled ‘duality’ has been defined for GrassmanHopf al/gebras. Thus,
an algebra not only has the usual reversed arrow dual coalgebra but a mirror (or tangled)
gebra which is quite distinct from the coalgebra.
Note: The dual of a quantum group^{} is a Hopf algebra^{}.
References
 1 W. Nichols and M. Sweedler, Hopf algebras and combinatorics^{}, in Proceedings of the conference on umbral calculus and Hopf algebras, ed. R. Morris, AMS, 1982.

2
B. Fauser: A treatise on quantum Clifford Algebras^{}. Konstanz,
Habilitationsschrift.
arXiv.math.QA/0202059 (2002).  3 B. Fauser: Grade Free product Formulae from Grassman–Hopf Gebras. Ch. 18 in R. Ablamowicz, Ed., Clifford Algebras: Applications to Mathematics, Physics and Engineering, Birkhäuser: Boston, Basel and Berlin, (2004).
 4 J. M. G. Fell.: The Dual Spaces of C*–Algebras., Transactions of the American Mathematical Society, 94: 365–403 (1960).
Title  dual of a coalgebra is an algebra, the 

Canonical name  DualOfACoalgebraIsAnAlgebraThe 
Date of creation  20130322 16:34:20 
Last modified on  20130322 16:34:20 
Owner  mps (409) 
Last modified by  mps (409) 
Numerical id  8 
Author  mps (409) 
Entry type  Derivation 
Classification  msc 16W30 
Related topic  GrassmanHopfAlgebrasAndTheirDualCoAlgebras 
Related topic  DualityInMathematics 
Related topic  QuantumGroups 
Defines  dualities of algebraic structures 