# Pointwise convergence of linear operators

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?

• Here's a hint: use the uniform boundedness principle – Aweygan Jul 3 '18 at 12:40
• 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.. – User1 Jul 3 '18 at 12:49
• 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? – Aweygan Jul 3 '18 at 22:15
• 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 :) – User1 Jul 5 '18 at 10:08
• You're welcome. Glad to help! – Aweygan Jul 5 '18 at 20:30