This is a follow-up question to a previous one:
linear independence in a dual pair.
The following is from the Topological Vector Spaces by Schaefer:
The corollary has been proven independently here. Could anyone help me understand how it could be derived directly from the theorem in the blue box?
Removing dependent $f_i$, we can WLOG assume that $f_i$ are independent for $i=1,\ldots,n$. Assume that $g$ is not a linear combination of $f_i$, that is $f_i$ and $g$ together are $n+1$ linear independent functionals. By the theorem, there exist $n+1$ vectors $x_i$ with the Kronecker's delta property. One of those vectors, $x_{n+1}$, is such that $f_i(x_{n+1})=\delta_{i,n+1}=0$, but $g(x_{n+1})=\delta_{n+1,n+1}=1$. It contradicts the assumption about the kernel inclusion.
