TITLE: Does first isomorphism theorem hold in the category of normed linear spaces?
QUESTION [7 upvotes]: Consider the category of normed linear spaces over $\mathbb{C}$ with bounded linear maps as morphisms. If $M\subset X$ is a subspace, then the quotient space $X/M$ has a map $\|x+M\|: = \inf_{y\in M}\|x-y\|_X$, which is a norm iff $M$ is closed in $X$ under the topology induced by the norm of $X$. 
Let $f:L\to M$ be bounded linear map between two normed linear spaces, it is easy to check that $\ker f$ is indeed closed, thus $L/\ker f$ is a normed linear space itself. Is it true that
$$L/\ker f\cong f(L)$$
i.e. there exist invertible bounded linear map whose inverse is also bounded?
I checked that the canonical vector space isomorphism $L/\ker f \to f(L)$ is indeed bounded, but I have trouble showing that the inverse is bounded, or equivalently, continuous. In other words, I want to show that 
$$\forall \epsilon>0. \exists \delta>0. \|f(l)-f(m)\|<\delta \Longrightarrow \|(l+\ker f)-(m+\ker f)\|<\epsilon$$
Is it true or is there a counterexample? If the statement is true, then what about the stronger statement: If $f:L\to M$ is an ismorphism of linear spaces that is bounded, then $f^{-1}$ is also bounded.

REPLY [3 votes]: The answer is yes for Banach spaces, using the Open mapping Theorem.