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I am currently taking a course in Differential Topology where the book we are using the one by Hirsch. I have had some troubles trying to prove that the weak topology makes the composition of maps, continuous. Up to now, I have been able to prove that, given a sub-basic neighborhood of $C^r(M,N)$, say $\mathcal{N}^r(h;(\phi,U),(\psi,V),K,\epsilon)$, there is a neighborhood of $C^r(P,N)\times C^r(M,P)$, $\mathcal{N}^r(g;(\chi,G),(\psi,V),\bar G,\epsilon'') \times\mathcal{N}^r(f;(\phi,U),(\psi,G),K,\epsilon')$ such that $g \circ f \in \mathcal{N}^r(h;(\phi,U),(\psi,V),K,\epsilon)$. But I have not been that lucky with the 'differential part'.

I think I reached, for $\bar f$ and $\bar g$, $$\| D^l(\phi \circ \bar g \circ \bar f \circ \phi^{-1}) - D^l(\phi \circ g \circ f \circ \psi^{-1}) \| \leq \|D^l (\phi \circ \bar g \circ \chi^{-1}) \| \|D^l(\chi \circ \bar f \circ \phi^{-1}) - D^l(\chi \circ f \circ \phi^{-1})\| + \|D^l (\phi \circ f \circ \chi^{-1}) \| \|D^l(\chi \circ \bar g \circ \phi^{-1}) - D^l(\chi \circ g \circ \phi^{-1})\|,$$ but I can not find a way to conclude it. Any ideas?

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