$\|T_{2}\|<\infty \implies \|T_2x\|_{Y} \leq C'' \|x\|_{X}$

Question

Let $X$ and $Y$ be a Banach spaces. Let $T_1:X\to Y$ be a bounded linear operator with $\|T_1x\|_{Y} \leq C \|x\|_{X}$ for some constant $C.$ Let $T_{2}:X\to Y$ be a linear operator with the property: $\|T_{2}\| \leq C' \|T_{1}\|$ for some constant $C'.$

(Here $\|T_i\|$ denotes the operator norm of $T_i$, and $\|\cdot \|_{X}$ denotes the norm of element in Banach space $X$)

Question:

Can we find some constant $C''$ such that $\|T_2x\|_{Y} \leq C'' \|x\|_{X}$?

Answer 1

From $\|T_{2}\| \leq C' \|T_{1}\|$ we get

$$\|T_2x\|_{Y} \leq ||T_2||||x||_X \leq C' \|T_{1}\|\|x\|_{X}$$

Hence you can take $C''=C' \|T_{1}\|$