Product of three bounded linear operators

Question

Let $A,B,C:X\rightarrow Y$ be bounded and (of course linear) operators. I want to examine if their product is also linear and bounded. What I have done:

\begin{align} ABC(\lambda x + \mu y)&=AB[\lambda C x + \mu C y ]=A[\lambda (BC) x + \mu (BC) y ]\\ &=\lambda (ABC) x + \mu (ABC) y \end{align} since the operators are linear. Furthermore, \begin{align} ||(ABCx)||=\|A(BCx)\| \leq K_1 \|B(Cx)\| \leq K_1 K_2\|Cx\| \leq K_1K_2K_3 \|x\| = K^{'}||x|| \end{align} proving that their product is bounded.

Am I correct?

Answer 1

You seem to have the right ideas, but you have a significant typo in your second set of equations. You should instead write something like $$ \|A(BCx)\| \leq K_1 \|B(Cx)\| \leq K_1 K_2\|Cx\| \leq K_1K_2K_3 \|x\| $$ Otherwise, you're correct.