a question about functional analysis Could you please help me to prove the following:

Let $f$ be any non-zero functional on a vector space $X$ and $x_0$ is a fixed element of $X\setminus N(f)$, where $N(f)$ is the kernel of $f$. Show that any $x\in X$ has a unique representation $x = ax_0 + y$, $y\in N(f)$.


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*If $x\in X$ then $x-\dfrac{f(x)}{f(x_0)}x_0\in N(f)$ (why?)...

*If $x = a_1x_0 + y_1= a_2x_0 + y_2$ then $ (a_1-a_2)x_0 + (y_1-y_2)\in N(f)$ ... since $(ax_0\in N(f)\iff a=0) \Longrightarrow$ ...

