Let $V$ be a countable infinite dimensional linear space, $$ e_1, e_2, e_3, \ldots $$ are the basis of $V$. $$ f: V \rightarrow V $$ is a left invertible linear map. For any $e_i$, assume $$ f(e_i)= f_{i1}e_1 + f_{i2}e_2 + f_{i3}e_3 +\cdots. $$ If there is a positive integer $R$ such that for any $i\in \mathbb N$, the number of nonzero elements of $$ \{f_{ij}\}_{j=1}^\infty $$ and the number of nonzero elements of $$ \{f_{ji}\}_{j=1}^\infty $$ are less than or equal to $R$, then how to show there is a left inverse $f^{-L}$ of $f$ such that :
there is a positive integer $R'$ such that for any $i\in \mathbb N$, the number of nonzero elements of $$ \{f_{ij}^{-L}\}_{j=1}^\infty $$ and the number of nonzero elements of $$ \{f_{ji}^{-L}\}_{j=1}^\infty $$ are less than or equal to $R'$ ? Where $f_{ij}^{-L}$ is the coefficient of $f^{-L}$, namely, for any $e_i$, we assume $$ f^{-L}(e_i)= f_{i1}^{-L}e_1 + f_{i2}^{-L}e_2 + f_{i3}^{-L}e_3 +\cdots. $$
PS: The left inverse of $f:V\rightarrow V$ is a map $f^{-L}:V\rightarrow V$ such that $f^{-L}\circ f$ is identity mapping. Namely, for any $x\in V$, we have $f^{-L}\circ f(x)=x$. If such $f^{-L}$ exists, we say $f$ is left invertiable. Notice that the left inverse maybe not unique in the infinite dimensional spaces.
PS: This problem is my guess. I think some examples, although there are some $f$ has left inverse which not satisfy the condition, but for there $f$, the left inverse is not uniquess, and I can find the one satisfy the condition. Thus I sum up guess, but after spending much time, I fail to prove it. I really want to whether it is right? If right, how to prove? Thanks for any help.
Without further hypotheses the claim is false. Take a small perturbation of the identity like e.g. $f(e_i)=e_i- \epsilon e_{i+1}$, $i\geq 0$ and $\epsilon$ small but non-zero. This verifies the required conditions. Write $f={\bf 1} - A$ and develop $(1-A)^{-1}=1+A+A^2...$ Then $f^{-L}(e_i)=e_i+\epsilon e_{i+1} + \epsilon^2 e_{i+2} + ...$ converges (in any reasonable space) and does not have finite support in the indices.