I would like to prove the fundamental theorem of homomorphisms of Banach space.
Let $V$ and $W$ be Banach spaces.Let $f:V→W$ be surjective bounded linear map.
I could prove $T:V/ker f→W$(v modKerf→Tv) is continuous.
I understood $\ker f$ is closed in $V$, so $V/\ker f$ is a Banach space.
What I should do is to prove inverse of $T$ is continuous.
If I could prove $T$ is bijective, by using range theorem, I can finish.
Could you tell me the proof of $T$ is bijective?
If I could prove this, I would be able to prove a kind of analogy of the fundamental theorem of homomorphisms for Banach spaces. Thank you in advance.
Best Answer
You have the famous Open Mapping Theorem at your hand. By this, you know that $f:V\rightarrow W$ is an open map as it is surjective. Now by an elementary property of quotient maps of Banach spaces (only Normed linear space is enough though), openness of a bounded linear map is preserved under quotients, i.e. $T:V/ker f\rightarrow W$ is a bounded open surjective map.
Now you just need to show that $T$ is injective. This is very simple. If $T[v]=T[w]$ where $[v]$ is the equivalence class of $v\in V$ in $V/kerf$, you have $T[v-w]=0\implies f(v-w)=0 \implies v-w\in kerf$. Hence you have that $[v]=[w]$. Thus $T$ is injective. Thus T is an open continuous bijection, hence the inverse is continuous.