Some infinite dimensional linear algebra, kernels of linear maps I'm studying functional analysis (namely weak convergence) and need to prove the following result: if $f,f_1,\ldots f_n$ are some linear maps $X\to \mathbb{C}$, where $X$ is a vector space over $\mathbb{C}$ then the inclusion $\bigcap\mathrm{Ker}f_i\subset \mathrm{Ker}f$ implies that $f\in \mathrm{Span}(f_1,\ldots f_n)$. It can be easily seen in the case $n=1$: $\mathrm{Ker}f_1\subset \mathrm{Ker}f$ implies that $\mathrm{Ker}f_1=\mathrm{Ker}f$ and immediately $f_1=cf$ for some $c\in \mathbb{C}$. Could you help me with the general case?

The map $(f_1,\dots,f_n):V\to\mathbb C^n$ factors through the injective map $F:V/\bigcap\mathrm{Ker}f_i\to\mathbb C^n$. Since $\bigcap\mathrm{Ker}f_i\subset \mathrm{Ker}f$, the map $f:V\to\mathbb C$ factors through $\tilde f:V/\bigcap\mathrm{Ker}f_i\to\mathbb C$. Since $F$ is injective, we can extend the linear form $\tilde f$ to a linear form on $\mathbb C^n$ (i.e. $\tilde f$ is the composition $V/\bigcap\mathrm{Ker}f_i\to\mathbb C^n\to\mathbb C$). If that extended form sends the $i$-th basis vector of $\mathbb C^n$ to $c_i$ then $f=\sum_i c_i f_i$.
