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tions follow. 

Consider a space of well-behaved ”test” functions, D$_{∞}$ ( R k ) of infinitely differentiable functions with bounded support, or any of the spaces D$_{m}$ ( R k ) of m times continuously differentiable functions (with bounded support); sometimes the domain of definition can be an open subset W of R $^{k}$, typically here W = (0 , 1) $^{k}$. Denote the generic space by D ( W ) ; convergence in D ( W ) is defined as follows: a sequence ψ$_{n}$ ∈ D ( W ) converges to zero if all ψ$_{n}$ are defined on a common bounded support in W and ψ$_{n}$ as well as all the l − th order derivatives (with l ≤ m for D$_{m}$ or all l < ∞ for D$_{∞}$ ) converge pointwise to zero. The space of generalized functions is the dual space, D $^{∗}$, the space of linear continuous functionals on D ( W ) with the weak topology: a sequence of elements of D ∗ converges if the sequence of values of the functionals converges for any test function from D ( W ) . The usual notation is to write the value of the functional f applied to a test function ψ ∈ D ( W ) as ( f, ψ ) ; then a sequence f$_{n}$ converges to f if for any ψ convergence ( f$_{n}$, ψ ) → ( f, ψ ) holds. 

Assume that functions in D ( W ) ; W ⊆ R k are suitably differentiable , e.g. at least k times continuously differentiable. Then for any ψ ∈ D ( W ) , and F ∈ D ∗ define a generalized derivative f ∈ D $^{∗}$; f = ∂ k ∂x 1 $_{...∂x$_{k}$}$F as the functional with values given by: 

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If the right-hand side is expressed via a regular locally summable function 

