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implies that a generalized function considered as a functional can sometimes be extended to a linear continuous functional on a wider space. 

Note that D ⊂ S and thus there is the inclusion of the linear dual spaces: S ∗ ⊂ D $^{∗}$; convergence in S ∗ of linear continuous functionals implies their convergence in D $^{∗}$, however, a sequence of elements of S ∗ that converges in D ∗ may not converge in the topology of S ∗ (see the example in section 3.2). 

In the terminology of Schwartz (1966) generalized functions are sometimes called ”distributions” and elements of S ∗ ”tempered distributions”; here we shall call them generalized functions indicating the specific space considered. In Sobolev (1992, p.59) a diagram shows various chains of generalized func- tions spaces embedded in each other; these are spaces of functionals on spaces of continuously differentiable (of different orders) functions, continuously dif- ferentiable functions with compact support and Sobolev spaces. 

Any locally summable (integrable on any bounded set) function b ( t ) de- fines a generalized function b in D ∗ by 

on the space of real-valued test functions or by 

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for complex-valued functions. Any locally summable function b ( t ) that ad- 

