Extension of a linear functional from a kernel space

Question

Let $X$ be a normed space, $f_1^*, f^*_2 ... f^*_n, f \in X^*$, and $G=\{x\in X: f_i^*(x) = 0, i=1,2...n\}$

Let $g$ be the restriction of $f$ in $G$ ($g=f|_G$) Prove that any continuous linear extension of $g$ is of the form $f+\alpha_1f_1 + ... \alpha_nf_n$.

I can see that this is a valid extension, but I do not know how to begin to characterize all continuous linear extensions since Hahn-Banach only talks about the existence of extensions. Would love to get some help on how to get started on this.

Answer 1

If $\tilde g$ is an extension of $g$, then $(\tilde g-f)|_G=0$. This can be written in the form $$\tag1 \bigcap_{j=1}^n\ker f_j\subset\ker(f-\tilde g). $$ Now you want to show that $(1)$ implies that $$ f-\tilde g=\lambda_1f_1+\cdots+\lambda_nf_n. $$ You probably want to do it for $n=1$ first.