  Given a sequence of finite element spaces which form a de Rham sequence, we
will construct a dual representation of these spaces with associated
differential operators which connect these spaces such that they also form a de
Rham sequence. The matrix which converts primal representations to dual
representations -- the Hodge matrix -- is the mass or Gram matrix. It will be
shown that a bilinear form of a primal and a dual representation is equal to
the vector inner product of the expansion coefficients (degrees of freedom) of
both representations. This leads to very sparse system matrices, even for high
order methods. The derivative of dual representations will be defined. Vector
operations, grad, curl and div, for primal and dual representations are both
topological and do not depend on the metric, i.e. the size and shape of the
mesh or the order of the numerical method. Derivatives are evaluated by
applying sparse incidence and inclusion matrices to the expansion coefficients
of the representations. As illustration of the use of dual representations, the
method will be applied to i) a mixed formulation for the Poisson problem in 3D,
ii) it will be shown that this approach allows one to preserve the equivalence
between Dirichlet and Neumann problems in the finite dimensional setting and,
iii) the method will be applied to the approximation of grad-div eigenvalue
problem on affine and non-affine meshes.
