Problem on Normed Linear Space

Question

$T$ is a continuous linear transformatin of a normed linear space $N$ to a normed linear space $N'$, $M$ is it's null space, show that $T$ induces a natural linear transformation $T'$ of $N/M$ to $N'$ and $||T'||=||T||$

My Attempt: $T':N/M\rightarrow N'$ defined by $T'(x+M)=T(x)$, well defined. Now, $||x+M||=inf_{m\in M} \{||x+m|| :m\in M\}\leq ||x+0||=||x||$

$\therefore$ for a fixed $x\notin M, \frac{||Tx||}{||x+M||} \geq \frac{||Tx||}{||x||} \Rightarrow ||T'||\geq ||T||$

I am struggling with the other inequality $||T'||\leq ||T||$

Answer 1

Hint: Note that $$ T'(x+M) = T(x) = T(x+y) $$ for all $y\in M$.