Following question:

Let X,Z be normed spaces, Y a Banach-space. Let $S_k,S \in \mathcal{L}(X,Y), T_k,T \in \mathcal{L}(Y,Z)$ and $S_k \rightarrow S, T_k \rightarrow T$ pointwise.

Then $T_k \circ S_k \rightarrow T \circ S$ pointwise.

Can somebody help me with this?

  • 1
    $\begingroup$ Here's a hint: use the uniform boundedness principle $\endgroup$ – Aweygan Jul 3 '18 at 12:40
  • $\begingroup$ Hey, is this the only way? Because we didn't have the Banach-Steinhaus theorem in our functional analysis lecture yet so I can't use it without proof.. $\endgroup$ – User1 Jul 3 '18 at 12:49
  • $\begingroup$ Given that you think you cannot apply the UBP, can you provide some context? Is this from a book, lecture notes, homework, etc? If so, can you link to a pdf, and/or let us know what section this is from? $\endgroup$ – Aweygan Jul 3 '18 at 22:15
  • $\begingroup$ Hey, my fault, we didn't have the Banach-Steinhaus theorom but indeed we had the uniform boundedness principle. Using this, the proof ist pretty easy so thanks :) $\endgroup$ – User1 Jul 5 '18 at 10:08
  • $\begingroup$ You're welcome. Glad to help! $\endgroup$ – Aweygan Jul 5 '18 at 20:30

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