Confusion about Ascoli's formula

Question

I'm trying to prove Ascoli's formula: $$\mathrm{dist}(x,\ker f)=\frac{|f(x)|}{\|f\|}$$ But I struggle, because I have problems believing it holds.

Let's supposer we're in a 3D space with functional

$f=(2\ 1\ 0)$, $\ker f = \mathrm{span}\{(0\ 0\ 1)^T\}$, $\|f\|=2$

Now let's suppose my point in space is $x=(0\ 1\ 0)^T$, $f(x)=1$.

Now according to the theorem:

$\mathrm{dist}(x,\ker f)=\frac{1}{2}$,

but $\mathrm{dist}(x,\ker f)=1$

What am I doing wrong?

Answer 1

The kernel of a (non-zero) linear functional always has codimension $1$, so in this case the kernel must have dimension $2$. In fact, we have $$ \ker(f)=\mathrm{span}\{(1,-2,0)^T,(0,0,1)^T\} $$