Topological classification of strong duals to nuclear (LF)-spaces

We show that the strong dual X' to an infinite-dimensional nuclear (LF)-space is homeomorphic to one of the spaces: ${\displaystyle {ℝ}^{\omega }}$, ${\displaystyle {ℝ}^{\infty }}$, ${\displaystyle Q\times {ℝ}^{\infty }}$, ${\displaystyle {ℝ}^{\omega }\times {ℝ}^{\infty }}$, or ${\displaystyle {\left({ℝ}^{\infty }\right)}^{\omega }}$, where ${\displaystyle {ℝ}^{\infty }=lim{ℝ}^n}$ and ${\displaystyle Q={[-1,1]}^{\omega }}$. In particular, the Schwartz space D' of distributions is homeomorphic to ${\displaystyle {\left({ℝ}^{\infty }\right)}^{\omega }}$. As a by-product of the proof we deduce that each infinite-dimensional locally convex space which is a direct limit of metrizable compacta is homeomorphic either to ${\displaystyle {ℝ}^{\infty }}$ or to ${\displaystyle Q\times {ℝ}^{\infty }}$. In particular, the strong dual to any metrizable infinite-dimensional Montel space is homeomorphic either to ${\displaystyle {ℝ}^{\infty }}$ or to ${\displaystyle Q\times {ℝ}^{\infty }}$.