If $M$ is a closed linear subspace of a normed linear space $N$, and if $T$ is the natural mapping of $N$ onto $N/M$ defined by $T(x)$ = $x+M$, show that $T$ is a continuous linear transformation for which $||T||$ $\le 1$ .
This question is from GF Simmons's Introduction to Modern Analysis book. Now, I can prove why the mapping is linear but for other stuff, I have no clue. I am generally stumped when presented with problems involving quotient spaces. Also, how is $M$ being closed important?
Best Answer
The usual norm on the quotient space is $$ \Vert x + M \Vert_q := \inf_{m\in M} \Vert x + m \Vert. $$ Now you can use the definition of the operator norm to show $\Vert T \Vert_{op} \leq 1$. Indeed, we have $0\in M$ and thus $$ \Vert T x\Vert_q = \Vert x + M \Vert_q = \inf_{m\in M} \Vert x + m \Vert \leq \Vert x \Vert. $$ This implies $\Vert T \Vert_{op} \leq 1$.
As pointed out in my comment, we need $M$ to be closed for $\Vert \cdot \Vert_q$ to be a norm. Imagine for example you take a dense subspace, then infimum is always zero (ie it is not a norm, because it is not positive definite).