I've been having trouble with this problem (2.8, problem 9) from 'Introductory Functional Analysis with Applications' by Kreyszig.
"Let $f\neq0$ be any linear functional on a vector space X and $x_0$ any fixed element of $X-\mathbb{N}(f)$, where $\mathbb{N}(f)$ is the null space of $f$. Show that any $x\in X$ has a unique representation $x=ax_0+y$, where $y\in \mathbb{N}(f)$."
I just don't understand how this can be true. Let $X=\mathbb{R}^2$, and define the linear functional $f$ as just the usual norm on $\mathbb{R}^2$. Let $x_0=(1,0)$ and let $x=(0,1)$. Then no linear combination of the form $x-ax_0$ is equal to $(0,0)$ (which is the only element of the null space).
How is this argument wrong? I think I might have made an obvious mistake or I have misinterpreted some definition. Any help greatly appreciated.
