Nyet, and nyet. This is why conscientious authors define the terms they use. A tempered distribution is a linear functional on a space of differentiable functions, for example, D_x(f) = f'(x), the derivative of f at x. This is why tempered distributions cannot be composed.
In general, the dual of a space of functions is a space of set functions, aka measures. https://keithalewis.github.io/math/dual.html