The following is Exercise 6 page 72 in Functional Analysis book of Conway :
Let $M = {\{x \in l^p : x(2n) = 0 \ \text{for all n}}\}$, $1 \le p \le \infty$. Show that $l^p/M$ is isometrically isomorphic to $l^p$.
First of all, am I right that I need to show three things to prove "isometrically isomorphism" : Surjection, Injection and Isometry?
1- Surjection : because the map is the canonical map so it surjection by definition.
2- Injection : I doubt that this is a 1-1 map unless we deter/adjust the definition of 1-1 here (?)
3- Roughly speaking $\inf_{m \in M} \sum_n |x_n|^p = \sum_n |x_n|^p$ because the zero sequence belongs to $M$ so it is isometry?
Best Answer
It is clear that $M$ is a closed subset of $\ell_p$; this implies that $\ell_p/M$ is well defined as a Banach space with norm $$\|x+M\|=d(x,M)=\inf\{\|x-y\|_p:y\in\mathbb{M}\}$$ Let $\pi:\ell_p\rightarrow\ell_p/M$ be the quotient map $x\mapsto x+M$.
The operator $T:\ell_p\rightarrow\ell_p$ given by $$(Tx)(n)=x(2n),\qquad n\in\mathbb{N}$$ is bounded: $$\|Tx\|^p_p=\sum_n|x(2n)|^p\leq \sum_n|x(n)|^p=\|x\|^p_p$$
It is easy to check that for any $x\in \ell_p$, $\|x+M\|=\|Tx\|_p$, for if $y\in M$ $$\|x-y\|^p_p=\sum_n|x(2n)|^p+\sum_n|x(2n-1)-y(2n-1)|^p\geq\sum_n|x(2n)|^p_p$$
Since $Tx=Ty$ whenever $x-y\in M$, and $\pi$ is a projection map, there exists a unique map $g:\ell_p/M\rightarrow \ell_p$ such that $$ T=g\circ \pi$$ It is easy to check that $g$ is linear and surjective: $g(x+M)=Tx$. Furthermore $$\|g(x+M)\|_p=\|Tx\|_p=\|x+M\|$$
The map $g$ is an isometric isomorphism between $\ell_p/M$ and $\ell_p$.